A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.</p>
<ol class="i-list"><li data-list-item-id="gp-q2-i" data-list-item-path="gp-q2-i"><div class="list-item-paragraph">The first four possible winning sequences for Kai are $KK$, $LKK$, $KLKK$ and $LKLKK$, where&nbsp;$K$ denotes a round won by Kai and $L$ denotes a round won by Lina. Find the probability of each sequence.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="gp-q2-ii" data-list-item-path="gp-q2-ii"><div class="list-item-paragraph">Hence, using the sum to infinity of a geometric progression, find the probability that Kai wins the match.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="gp-q2-iii" data-list-item-path="gp-q2-iii"><div class="list-item-paragraph">Given that Lina wins the first round, find the probability that Kai wins the match.<span class="inline-marks">[2]</span></div></li></ol>

Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.

Conditional Probability

Repeated Trials with Infinitely Many Possible Outcomes

Probabilities of Successive Events

Application of Sum to Infinity

Waiting-Time and Stopping-Rule Probabilities

Sequence and Series

Probability

<p>Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.</p>
<ol class="i-list"><li data-list-item-id="gp-q1-i" data-list-item-path="gp-q1-i"><div class="list-item-paragraph">Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is $\dfrac{2}{5}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="gp-q1-ii" data-list-item-path="gp-q1-ii"><div class="list-item-paragraph">Find the probability that Ava wins the game.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.</p>
<ol class="i-list" start="3"><li data-list-item-id="gp-q1-iii" data-list-item-path="gp-q1-iii"><div class="list-item-paragraph">Find the probability that Ava wins the second game.<span class="inline-marks">[3]</span></div></li></ol>

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

Law of Total Probability

<p>A bag contains $10$ orange-flavoured, $14$ strawberry-flavoured, and $16$ cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet.</p>
<p>Calculate the probability that</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured,</span></div></li><li><div class="list-item-content"><span class="list-item-text">he selects an even number of sweets.</span></div></li></ol>

A bag contains $10$ orange-flavoured, $14$ strawberry-flavoured, and $16$ cherry-flavoured sweets which are of identical shapes and sizes. Benny selects a sweet at random from the bag. If it is not cherry-flavoured he replaces it and selects another sweet at random. He repeats the process until he obtains a cherry-flavoured sweet.

the first sweet selected is strawberry-flavoured and the fourth sweet is orange-flavoured,

<p>Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either Player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, Player $A$ has a probability of $p$ of winning this set.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">With the aid of a tree diagram, find in terms of $p$, the probability that player $A$ will win the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the range of values of the probability that player $A$ wins the match.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the value of $p$ in order for the match to be fair is $\frac{1}{8}$.</span></div></li></ol>
<p>Given that Player $A$ wins the match, find, in terms of $p$, the probability that he won the $2^{\text{nd}}$ set.</p>

Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either Player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, Player $A$ has a probability of $p$ of winning this set.

With the aid of a tree diagram, find in terms of $p$, the probability that player $A$ will win the match.

Deduce the range of values of the probability that player $A$ wins the match.

Show that the value of $p$ in order for the match to be fair is $\frac{1}{8}$.

Given that Player $A$ wins the match, find, in terms of $p$, the probability that he won the $2^{\text{nd}}$ set.

Finding Unknowns in Probability Models

2011 NYJC Prelim

<p>A shop sells plums in bags of $9$. Plums are taken from a large supply and the probability that a plum is rotten is $p$, where&nbsp;$0 &lt; p &lt; 0.2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">It is known that the variance of the number of rotten plums in a bag is $0.6624$. Find $p$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$p = 0.08$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a randomly selected bag of plums has at least $3$ rotten plums.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Five bags of plums are randomly selected. Find the probability that the fifth selected bag is the second bag with at least $3$ rotten plums.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The shop sells $30$ bags of plums to a customer. Find the probability that less than $90$% of the bags have fewer than $3$ rotten plums.</span><span class="inline-marks">[3]</span></div></li></ol>

A shop sells plums in bags of $9$. Plums are taken from a large supply and the probability that a plum is rotten is $p$, where $0 < p < 0.2$.

It is known that the variance of the number of rotten plums in a bag is $0.6624$. Find $p$.

Find the probability that a randomly selected bag of plums has at least $3$ rotten plums.

Five bags of plums are randomly selected. Find the probability that the fifth selected bag is the second bag with at least $3$ rotten plums.

The shop sells $30$ bags of plums to a customer. Find the probability that less than $90$% of the bags have fewer than $3$ rotten plums.

2024 YIJC H1 Q9

Discrete Random Variables (DRV)

<p>One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of hand phones ($x$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of students ($f$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$fx$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$15$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$c$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$50$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$b$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$d$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$a$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$84$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Total</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$100$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$e$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the values of $a$, $b$, $c$, $d$ and $e$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the mean and standard deviation of the number of hand phones in their home.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">There are $5$ red cards, $6$ yellow cards and $4$ blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Brandon draws $2$ cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.</span></div></li></ol>

One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.

There are $5$ red cards, $6$ yellow cards and $4$ blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.

Brandon draws $2$ cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.

Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.

Mean and Standard Deviation for Ungrouped Data

Constructing and Interpreting Frequency Tables

Probability of a Single Event

Statistics

<p>A bag contains $4$ identical red balls and $6$ identical blue balls.</p>
<ol class="abc-list"><li data-list-item-id="item-a-5" data-list-item-path="item-a-5"><div class="list-item-paragraph">In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.</div><ol class="i-list"><li data-list-item-id="item-a-5-i" data-list-item-path="item-a-5/item-a-5-i"><div class="list-item-paragraph">Find the probability that the last ball picked is red.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-5-ii" data-list-item-path="item-a-5/item-a-5-ii"><div class="list-item-paragraph">Find the probability that in the first five picks, exactly $2$ blue balls are picked given that at least $3$ red balls are picked.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b-5" data-list-item-path="item-b-5"><div class="list-item-paragraph">In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of $20$ random picks from the bag and the colour of each pick is recorded.</div><ol class="i-list"><li data-list-item-id="item-b-5-i" data-list-item-path="item-b-5/item-b-5-i"><div class="list-item-paragraph">Find the probability that the colour red is recorded exactly $4$ times.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-ii" data-list-item-path="item-b-5/item-b-5-ii"><div class="list-item-paragraph">Find the probability that the colour red is recorded more than $4$ times but not more than $8$ times.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-iii" data-list-item-path="item-b-5/item-b-5-iii"><div class="list-item-paragraph"><p>Find the probability that the $8 ^\mathrm{th}$ pick is the $6^\mathrm{th}$ time the colour blue is recorded.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-5-iv" data-list-item-path="item-b-5/item-b-5-iv"><div class="list-item-paragraph">The second game is played by $50$ randomly chosen players. Estimate the probability that the average number of times the colour red is recorded is more than $8.5$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

A bag contains $4$ identical red balls and $6$ identical blue balls.

In the first game, the balls are randomly picked by a player from the bag, one at a time, without replacement, until there are no balls left in the bag.

In the second game, each ball picked by a player will be placed back into the bag before the next ball is picked. The player makes a total of $20$ random picks from the bag and the colour of each pick is recorded.

Central Limit Theorem

Selections with Restrictions

2025 RI P2 Q10

Sampling

Permutation and Combination

<p></p>
<p>A bag contains $4$ black balls and $6$ white balls. Three balls are drawn from the bag at random, one by one. If a black ball is drawn, it is replaced in the bag. If a white ball is drawn, it is not replaced and $2$ extra black balls are added to the bag.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that all balls drawn are white is $\frac{1}{11}$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that at least one of the balls drawn is black, find the probability that all balls drawn are black.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A fourth ball is further drawn from the bag at random.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the third white ball appears on this fourth draw.</span><span class="inline-marks">[3]</span></div></li></ol>

A bag contains $4$ black balls and $6$ white balls. Three balls are drawn from the bag at random, one by one. If a black ball is drawn, it is replaced in the bag. If a white ball is drawn, it is not replaced and $2$ extra black balls are added to the bag.

Show that the probability that all balls drawn are white is $\frac{1}{11}$.

Given that at least one of the balls drawn is black, find the probability that all balls drawn are black.

A fourth ball is further drawn from the bag at random.

Find the probability that the third white ball appears on this fourth draw.

Constructing and Using Probability Trees

Conditional Probability from Tree Diagrams

2013 JJC P2 Q9

<p>A bag contains $15$ tokens that are indistinguishable apart from their colours. $2$ of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the probability that a participant draws $2$ red tokens on the first $2$ draws is $\frac{1}{35}$, show that there are $3$ red tokens in the bag.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the probability that a participant draws a red or green token on the second draw.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Events $A$ and $B$ are defined as follows:</div><div class="list-item-paragraph">$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $A$: A participant draws his/her second red token on the third draw.<br>$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $B$: A participant draws a blue token on the second draw.</div><div class="list-item-paragraph">Find $\mathrm{P}(A \cup B)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine if&nbsp;$A$ and $B$ are independent events.<span class="inline-marks">[1]</span></div></li></ol>

A bag contains $15$ tokens that are indistinguishable apart from their colours. $2$ of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.

Given that the probability that a participant draws $2$ red tokens on the first $2$ draws is $\frac{1}{35}$, show that there are $3$ red tokens in the bag.

Find the probability that a participant draws a red or green token on the second draw.

Independent Events

Probability Using Permutations and Combinations

Addition Rule for Overlapping Events

2013 HCI P2 Q10

<p>A basketball free throw game involves a team of two players, Ben and John. The game consists of at most $3$ throws and the moment a player makes a successful shot, the team wins, and the game will end.</p>
<p>The game uses the following rules.</p>
<ul class="bullet-list" style="margin-left: 2.5rem"><li data-list-item-id="rule-1" data-list-item-path="rule-1"><div class="list-item-paragraph">Only one player is selected for each game and the probability that Ben is selected in a game is $0.7$.</div></li><li data-list-item-id="rule-2" data-list-item-path="rule-2"><div class="list-item-paragraph">The probabilities that Ben and John make a successful shot in any single attempt are $0.1$ and $0.07$ respectively.</div></li><li data-list-item-id="rule-3" data-list-item-path="rule-3"><div class="list-item-paragraph">The shots are independent of each other.</div></li></ul>
<ol class="abc-list"><li data-list-item-id="item-a-4" data-list-item-path="item-a-4"><div class="list-item-paragraph">Find the probability that the team wins the game.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-4" data-list-item-path="item-b-4"><div class="list-item-paragraph">The team did not win the game. Find the probability that John was the one who was selected to shoot.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c-4" data-list-item-path="item-c-4"><div class="list-item-paragraph">The team attempts the game repeatedly until the first game is won. Find the least number of attempts required such that the probability of winning within $n$ games is at least $0.95$.<span class="inline-marks">[2]</span></div></li></ol>

A basketball free throw game involves a team of two players, Ben and John. The game consists of at most $3$ throws and the moment a player makes a successful shot, the team wins, and the game will end.

Only one player is selected for each game and the probability that Ben is selected in a game is $0.7$.

The probabilities that Ben and John make a successful shot in any single attempt are $0.1$ and $0.07$ respectively.

Find the probability that the team wins the game.

The team did not win the game. Find the probability that John was the one who was selected to shoot.

The team attempts the game repeatedly until the first game is won. Find the least number of attempts required such that the probability of winning within $n$ games is at least $0.95$.

2025 HCI P2 Q6

<p>At a shopping mall, one ticket is given to every customer who spends $\$1000$ or more. With the ticket, the customer will be invited to draw a ball from the lucky draw box containing $35$ white and $5$ orange balls.</p>
<p>If the first draw is an orange ball, the customer wins a special prize. The draw ends and the ball is not replaced.</p>
<p>If the first draw is a white ball, the ball is replaced in the box and the customer is invited to draw a second time.</p>
<p>If the second draw is an orange ball, the customer wins the special prize. The draw ends and the ball is not replaced.</p>
<p>If the second draw is a white ball, the ball is replaced in the box and customer walks away with nothing.</p>
<p>The lucky draw is over when all the $5$ orange balls have been drawn by customers.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">If none of the special prizes have been won yet, find the probability that the customer with the ticket</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">draws $2$ white balls,</div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">wins the special prize.</div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">If only one orange ball is left, find the probability of winning the special prize.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">By considering your answers and the format of the lucky draw, will it take the next customer a longer/shorter/same amount of time to win the special prize?<br>Briefly explain your answer.</div></li></ol>

At a shopping mall, one ticket is given to every customer who spends $\$1000$ or more. With the ticket, the customer will be invited to draw a ball from the lucky draw box containing $35$ white and $5$ orange balls.

If the first draw is an orange ball, the customer wins a special prize. The draw ends and the ball is not replaced.

If the first draw is a white ball, the ball is replaced in the box and the customer is invited to draw a second time.

If the second draw is an orange ball, the customer wins the special prize. The draw ends and the ball is not replaced.

If the second draw is a white ball, the ball is replaced in the box and customer walks away with nothing.

The lucky draw is over when all the $5$ orange balls have been drawn by customers.

If none of the special prizes have been won yet, find the probability that the customer with the ticket

If only one orange ball is left, find the probability of winning the special prize.

Complementary Events

Dependent Events

Probability in a Two-Draw Lucky Dip

<p>A bag contains $10$ identical balls of which $4$ are red and $6$ are blue. Peter and Glen take turns to draw a ball from the bag without replacement. The first student to take a blue ball wins the game and the game ends.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that Peter takes the first draw, Glen takes the second draw, Peter then takes the third draw and so on, state, as a fraction, the probability that</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Peter wins at the first draw.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Glen wins at the second draw.</span></div></li><li><div class="list-item-content"><span class="list-item-text">no fourth draw is required.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">If the game ends at the $n^{\mathrm{th}}$ draw, state the largest value of $n$.</span></div></li></ol>

A bag contains $10$ identical balls of which $4$ are red and $6$ are blue. Peter and Glen take turns to draw a ball from the bag without replacement. The first student to take a blue ball wins the game and the game ends.

Given that Peter takes the first draw, Glen takes the second draw, Peter then takes the third draw and so on, state, as a fraction, the probability that

If the game ends at the $n^{\mathrm{th}}$ draw, state the largest value of $n$.

Sampling Without Replacement with a Stopping Rule

<p>There are $2$ bags, $A$ and $B$. $A$ contains $3$ red balls and $5$ green balls while $B$ contains $4$ red balls and $3$ green balls. A game is played by $2$ players where a fair $6$ sided die is first thrown.<br></p>
<p>If the die shows the numbers $1$ or $2$, the player picks a ball from Bag $A$ and places it in Bag $B$. He then picks a ball from Bag $B$. If the die shows $3$, $4$, $5$ or $6$, the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag $B$ wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.</p>
<p>The game stops when a player has won.</p>
<p>Find the probability that</p>
<ol class="abc-list"><li data-list-item-id="item-1766239416643-9a8qfpxv4" data-list-item-path="item-1766239416643-9a8qfpxv4"><div class="list-item-paragraph">Mark wins the game on the first try.</div></li><li data-list-item-id="item-1766239417625-iuw4tlw34" data-list-item-path="item-1766239417625-iuw4tlw34"><div class="list-item-paragraph">Anthony wins the game on his first try.</div></li></ol>
<p>Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.</p>

There are $2$ bags, $A$ and $B$. $A$ contains $3$ red balls and $5$ green balls while $B$ contains $4$ red balls and $3$ green balls. A game is played by $2$ players where a fair $6$ sided die is first thrown.


If the die shows the numbers $1$ or $2$, the player picks a ball from Bag $A$ and places it in Bag $B$. He then picks a ball from Bag $B$. If the die shows $3$, $4$, $5$ or $6$, the player loses his turn. The other player will then throw the die. The player who picks a red ball from Bag $B$ wins the game. Mark and Anthony take turns to play the game with Mark starting first. After each person has played the game, the balls are placed back into their original bags and the game starts again.

Every ball selection is uniform at random. Each die throw is independent, and the bags are reset before the next turn if nobody has won.

2020 ACSI EM S4 P1

<p>$65\%$ of the customers in a queue to a PC fair are males. </p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that there first three customers are females and next three are males.</div></li><li><div class="list-item-paragraph">Find the probability that the $4^{\mathrm{th}}$ male customer is the $7^{\mathrm{th}}$ customer in the queue.</div></li><li><div class="list-item-paragraph">There are $20$ customers in the queue. Find probability that there are more than $9$ male customers and the $4^{\mathrm{th}}$ male customer is the $7^{\mathrm{th}}$ customer in the queue.</div></li></ol>
<p>There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer is a female is greater than $0.995$. Find the minimum value of $n$.</p>

$65\%$ of the customers in a queue to a PC fair are males. 

There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer is a female is greater than $0.995$. Find the minimum value of $n$.

<p>Customers wait in a queue in a supermarket. A customer can choose to pay with cash or a credit card. Assume that whether or not a customer pays with a credit card is independent of any other customers’ methods of payment. It is known that $60\%$ of customers choose to pay with a credit card.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the probability that the first three customers pay with a credit card and the next three pay with cash;</div></li><li><div class="list-item-paragraph">Find the probability that the $4^{\mathrm{th}}$ person who pays with credit card is the $6^{\mathrm{th}}$ customer in the queue.</div></li><li><div class="list-item-paragraph">There are $15$ customers in the queue. Find the probability that there are more than $10$ customers who pay with a credit card and the $4^{\mathrm{th}}$ person who pays with credit card is the $6^{\mathrm{th}}$ customer in the queue.</div></li></ol>
<p>There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer pays with cash is greater than $0.995$.<br></p>
<p>Find the minimum value of $n$.</p>

Customers wait in a queue in a supermarket. A customer can choose to pay with cash or a credit card. Assume that whether or not a customer pays with a credit card is independent of any other customers’ methods of payment. It is known that $60\%$ of customers choose to pay with a credit card.

There are $n$ customers waiting in another queue in the same supermarket. The probability that at least one customer pays with cash is greater than $0.995$.


<p>Ava and Barry play a game containing one green marble and two red marbles. Each player in turn randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that Ava wins on her first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Barry wins on his first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Ava wins in one of her first three turns.</div></li><li><div class="list-item-paragraph">Find the probability that Ava eventually wins.</div></li></ol>

Ava and Barry play a game containing one green marble and two red marbles. Each player in turn randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.

<p>Ava and Barry play a game containing one green marble and two red marbles. Each player intron randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the probability that Ava wins on her first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Barry wins on his first turn.</div></li><li><div class="list-item-paragraph">Find the probability that Ava wins in one of her first three turns.</div></li><li><div class="list-item-paragraph">Find the probability that Ava eventually wins.</div></li></ol>

Ava and Barry play a game containing one green marble and two red marbles. Each player intron randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.

Worked Example 5

<p>A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that&nbsp;$1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.</p>
<ol class="i-list"><li data-list-item-id="item-1761026739916-r8mf46f1p" data-list-item-path="item-1761026739916-r8mf46f1p"><div class="list-item-paragraph">Show that the probability that a randomly selected packet is unsatisfactory is $0.0872$, correct to $3$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>
<p>The number of packets that are unsatisfactory is denoted by $X$. You may assume that&nbsp;$X$ can be modelled by a binomial distribution</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1761026778901-9zeslxr7m" data-list-item-path="item-1761026778901-9zeslxr7m"><div class="list-item-paragraph">Find the probability that there are at least $5$ but less than $10$ packets that are unsatisfactory.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026779456-k1lne558d" data-list-item-path="item-1761026779456-k1lne558d"><div class="list-item-paragraph">Find the least value of $r$ such that the probability that there are more than $r$ packets that are unsatisfactory is at most $0.12$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1761026803727-iw5hgizpy" data-list-item-path="item-1761026803727-iw5hgizpy"><div class="list-item-paragraph">How should the packets be selected? Give a reason for this method of selection.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761026804701-qwibu9mih" data-list-item-path="item-1761026804701-qwibu9mih"><div class="list-item-paragraph">Find the probability that the ninth packet is the third unsatisfactory packet selected.<span class="inline-marks">[2]</span></div></li></ol>

A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that $1\%$ of the notebooks, $2\%$ of the pens, and $4\%$ of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in $180$ packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.

The number of packets that are unsatisfactory is denoted by $X$. You may assume that $X$ can be modelled by a binomial distribution

Before selling the packets of stationery, he decided to select a sample of $9$ packets to check for unsatisfactory packets.

Binomial Probabilities Using Technology

Binomial Probabilities Using the Probability Mass Function

Population and Random Samples

2024 JPJC P2 Q8

<p>National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of $25$. It is known that $100p\%$ of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is $1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain why the context above may not be well-modelled by a binomial distribution.</div><p style="text-align: right;">[1]</p></li></ol>
<p>Assume now that the context above is well-modelled by a binomial distribution.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State the value of $p$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the probability that a box chosen at random has less than $2$ rotten tomato.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">A customer chose a box and inspected the contents individually. Find the probability that the twenty-first tomato is the fourth rotten tomato and no rotten tomatoes are found subsequently.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Boxes that contains at least $24$ tomatoes that are not rotten are deemed satisfactory.</p>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">A customer first picks $3$ boxes of tomatoes, of which at least $2$ boxes are satisfactory. The customer then decides to buy another $5$ boxes. Find the probability that exactly $6$ of the $8$ boxes are satisfactory.</div><p style="text-align: right;">[4]</p></li></ol>

National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of $25$. It is known that $100p\%$ of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is $1$.

Assume now that the context above is well-modelled by a binomial distribution.

Boxes that contains at least $24$ tomatoes that are not rotten are deemed satisfactory.

Binomial Models and Assumptions

Repeated Binomial Experiments

Determining an Unknown Binomial Success Probability

Mean and Variance of a Binomial Distribution

2021 NJC P2 Q8

<p>Amil and Babs are playing a game in which they take turns to throw a fair six-sided die. The winner is the first player to throw the same number as the other player has thrown immediately before. Anil throws first.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find exactly the probability that Anil wins the game.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that Babs wins the game, find exactly the probability that she wins on her second throw.</div><p style="text-align: right;">[3]</p></li></ol>

Amil and Babs are playing a game in which they take turns to throw a fair six-sided die. The winner is the first player to throw the same number as the other player has thrown immediately before. Anil throws first.

Math Question Bank