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<p>A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$p = 25$, find the probability that a randomly chosen component is faulty.</span></div></li><li><div class="list-item-paragraph">For a general value of $p$, the probability that a randomly chosen component that is faulty was supplied by $A$ is denoted by $\mathrm{f}(p)$. Show that&nbsp;$\mathrm{f}(p) = \frac{0.05p}{0.02p + 3}$. Prove by differentiation that&nbsp;$\mathrm{f}$ is an increasing function for&nbsp;$0 \leq p \leq 100$, and explain what this statement means in the context of the question.</div></li></ol>

A company buys $p\%$ of its electronic components from supplier $A$ and the remaining $(100 - p)\%$ from supplier $B$. The probability that a randomly chosen component supplied by $A$ is faulty is $0.05$. The probability that a randomly chosen component supplied by $B$ is faulty is $0.03$.

Given that $p = 25$, find the probability that a randomly chosen component is faulty.

For a general value of $p$, the probability that a randomly chosen component that is faulty was supplied by $A$ is denoted by $f(p)$. Show that $f(p) = \frac{0.05p}{0.02p + 3}$. Prove by differentiation that $f$ is an increasing function for $0 \leq p \leq 100$, and explain what this statement means in the context of the question.

N2009 P2 Q7

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.

2011 NYJC Prelim

<p>Box $A$ contains $6$ balls numbered $1, 2, 2, 2, 5, 7$.</p>
<p>Box $B$ contains $4$ balls numbered $1, 4, 4, 7$.</p>
<p>Box $C$ contains $3$ balls numbered $3, 4, 6$.</p>
<p>One ball is removed at random from each box.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the probability that each of the three numbers obtained is greater than $3$ is $\frac{1}{6}$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that each of the three numbers obtained is greater than $3$ or the sum of the three numbers obtained is $13$ (or both).</span><span class="inline-marks">[4]</span></div></li></ol>

Box $A$ contains $6$ balls numbered $1, 2, 2, 2, 5, 7$.

Box $B$ contains $4$ balls numbered $1, 4, 4, 7$.

Box $C$ contains $3$ balls numbered $3, 4, 6$.

One ball is removed at random from each box.

Show that the probability that each of the three numbers obtained is greater than $3$ is $\frac{1}{6}$.

Find the probability that each of the three numbers obtained is greater than $3$ or the sum of the three numbers obtained is $13$ (or both).

<p>Two events $A$ and $B$ are such that&nbsp;$\mathrm{P}(A) = p$ and $\mathrm{P}(B) = \frac{5}{4}p$. It is given that&nbsp;$\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $p$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether the events $A'$ and $B$ are independent.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The events $C$ and $D$ are such that&nbsp;$\mathrm{P}(C) = 0.55$, $\mathrm{P}(D) = 0.6$ and $\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the value of $q$ that gives the minimum value of $\mathrm{P}(A \cup C \cup D)$ and state the minimum value of $\mathrm{P}(A \cup C \cup D)$.</span><span class="inline-marks">[2]</span></div></li></ol>

Two events $A$ and $B$ are such that $\mathrm{P}(A) = p$ and $\mathrm{P}(B) = \frac{5}{4}p$. It is given that $\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6$.

Explain whether the events $A'$ and $B$ are independent.

The events $C$ and $D$ are such that $\mathrm{P}(C) = 0.55$, $\mathrm{P}(D) = 0.6$ and $\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q$.

Find the value of $q$ that gives the minimum value of $\mathrm{P}(A \cup C \cup D)$ and state the minimum value of $\mathrm{P}(A \cup C \cup D)$.

2025 VJC P2 Q6

<p>For events $A$, $B$ and $C$, it is given that&nbsp;$\mathrm{P}(A) = 0.3$, $\mathrm{P}(B) = 0.4$, $\mathrm{P}(C) = 0.42$, $\mathrm{P}(A' \cap B \cap C) = x$ and $\mathrm{P}(A \cap B \cap C) = 0.02$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A \cap C)$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of $x$ where necessary.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the greatest and least possible values of $\mathrm{P}(A' \cap B' \cap C)$.</span><span class="inline-marks">[2]</span></div></li></ol>

For events $A$, $B$ and $C$, it is given that $\mathrm{P}(A) = 0.3$, $\mathrm{P}(B) = 0.4$, $\mathrm{P}(C) = 0.42$, $\mathrm{P}(A' \cap B \cap C) = x$ and $\mathrm{P}(A \cap B \cap C) = 0.02$. It is also given that events $A$ and $B$ are independent, and that events $A$ and $C$ are independent.

Find $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A \cap C)$.

Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of $x$ where necessary.

Find the greatest and least possible values of $\mathrm{P}(A' \cap B' \cap C)$.

2025 TMJC P2 Q6

<p>In a simple archery game, a player shoots an arrow at a target. The probability of hitting the target depends on the player’s skill and the wind conditions for the day. The wind can either be good or bad, with the probability of a good wind day being $0.7$.</p>
<p>According to past statistics:</p>
<ul class="bullet-list"><li><div class="list-item-content"><span class="list-item-text">Dave has a $70$% chance of hitting the target on a good wind day and a $60$% chance on a bad wind day.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Rafael has an $80$% chance of hitting the target on a good wind day and a $50$% chance on a bad wind day.</span></div></li></ul>
<p>Dave and Rafael decide to compete in a game where each player shoots $2$ arrows at the target. The player who hits the target more times wins the game.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that Dave wins the game on a good wind day.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is now known that the probability Dave wins the game on a bad wind day is $0.39$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Given that Dave wins the game, find the probability that the day was a bad wind day.</span><span class="inline-marks">[3]</span></div></li></ol>

In a simple archery game, a player shoots an arrow at a target. The probability of hitting the target depends on the player’s skill and the wind conditions for the day. The wind can either be good or bad, with the probability of a good wind day being $0.7$.

Dave has a $70$% chance of hitting the target on a good wind day and a $60$% chance on a bad wind day.

Rafael has an $80$% chance of hitting the target on a good wind day and a $50$% chance on a bad wind day.

Dave and Rafael decide to compete in a game where each player shoots $2$ arrows at the target. The player who hits the target more times wins the game.

Find the probability that Dave wins the game on a good wind day.

It is now known that the probability Dave wins the game on a bad wind day is $0.39$.

Given that Dave wins the game, find the probability that the day was a bad wind day.

2025 TJC P2 Q5

<p>A game at a funfair involves pulling balls of different colours out of a bag. A bag contains $1$ white ball, $2$ blue balls, $3$ green balls and $4$ red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that a player draws a white ball and two green balls on their three attempts.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Events $A$ and $B$ are defined as</p>
<p style="margin-left: 14rem">$A$: A white ball is drawn on the first attempt<br>$B$: A white ball is drawn on the second attempt</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Determine with reason whether events $A$ and $B$ are independent.</span><span class="inline-marks">[2]</span></div></li></ol>

A game at a funfair involves pulling balls of different colours out of a bag. A bag contains $1$ white ball, $2$ blue balls, $3$ green balls and $4$ red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag.

Find the probability that a player draws a white ball and two green balls on their three attempts.

Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt.

$A$: A white ball is drawn on the first attempt
$B$: A white ball is drawn on the second attempt

Determine with reason whether events $A$ and $B$ are independent.

2025 RVHS P2 Q8

<p>A bag contains $4$ identical red balls and $6$ identical blue balls.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">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.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the last ball picked is red.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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><span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the probability that the colour red is recorded exactly $4$ times.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the colour red is recorded more than $4$ times but not more than $8$ times.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that the $8$th pick is the $6$th time the colour blue is recorded.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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><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.

2025 RI P2 Q10

Binomial Distribution

<p>Two fair six-sided dice are thrown. Events $A$, $B$ and $C$ are defined as follows.</p>
<p style="margin-left: 4rem">$A$: sum of the two scores is odd<br>$B$: at least one of the two scores is greater than $4$<br>$C$: the two scores are equal</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find, giving your reasons clearly in each case, which pair of the events are</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">mutually exclusive,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">independent.</span><span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}(C \mid B)$.</span><span class="inline-marks">[2]</span></div></li></ol>

Two fair six-sided dice are thrown. Events $A$, $B$ and $C$ are defined as follows.

$A$: sum of the two scores is odd
$B$: at least one of the two scores is greater than $4$
$C$: the two scores are equal

Find, giving your reasons clearly in each case, which pair of the events are

2025 NYJC P2 Q6

<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.

Probability - General

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.

2013 HCI P2 Q10

<p>A box contains $5$ red balls, $5$ green balls and $5$ blue balls. The balls are identical in size. Balls of the same colour are each printed with a distinct number from $1$, $2$, $3$, $4$, $5$. Two balls are randomly drawn without replacement. Find the probability that</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">at least $1$ red ball is drawn,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the numbers on the balls drawn is at least $9$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the numbers on the balls drawn is at least $9$ given that at least $1$ red ball is drawn.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>State, with a reason, whether the events 'the sum of the numbers on the balls drawn is at least $9$' and 'at least $1$ red ball is drawn' are independent.<span class="inline-marks">[1]</span></p>

A box contains $5$ red balls, $5$ green balls and $5$ blue balls. The balls are identical in size. Balls of the same colour are each printed with a distinct number from $1$, $2$, $3$, $4$, $5$. Two balls are randomly drawn without replacement. Find the probability that

the sum of the numbers on the balls drawn is at least $9$,

the sum of the numbers on the balls drawn is at least $9$ given that at least $1$ red ball is drawn.

State, with a reason, whether the events 'the sum of the numbers on the balls drawn is at least $9$' and 'at least $1$ red ball is drawn' are independent.

2013 DHS P2 Q7

<p>In a box, there are $12$ marbles of which $5$ are red, $4$ are black and $3$ are white. $4$ marbles are taken at random from the box, each marble being replaced before the next one is taken. Find the probability that</p>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">at least one marble of each colour is obtained,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">at least one marble is white.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>Everyday, Gerald has a choice of $2$ routes to get to school. The probability that he gets to school without being delayed is $\frac{43}{50}$. The probability that he chooses the first route is $\frac{3}{5}$. If he gets to school without being delayed, the probability that the first route is chosen is $\frac{27}{43}$. Find the probability that he gets to school without being delayed if</p>
<ol class="abc-list" start="2"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">he chooses the first route,</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">he chooses the second route.</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>

In a box, there are $12$ marbles of which $5$ are red, $4$ are black and $3$ are white. $4$ marbles are taken at random from the box, each marble being replaced before the next one is taken. Find the probability that

Everyday, Gerald has a choice of $2$ routes to get to school. The probability that he gets to school without being delayed is $\frac{43}{50}$. The probability that he chooses the first route is $\frac{3}{5}$. If he gets to school without being delayed, the probability that the first route is chosen is $\frac{27}{43}$. Find the probability that he gets to school without being delayed if

2013 TJC P2 Q10

<p>Events $A$ and $B$ are such that&nbsp;$\text{P}\left( A \right)=0.7$and $\text{P}\left( B \right)=0.8$.  </p>
<p>Find an inequality for&nbsp;$\text{P}\left( A\cap B \right)$.</p>

Events $A$ and $B$ are such that $\text{P}\left( A \right)=0.7$and $\text{P}\left( B \right)=0.8$.  

Find an inequality for $\text{P}\left( A\cap B \right)$.

<p>Events $A$ and $B$ are such that&nbsp;$\mathrm{P}\left( A \right)=0.74$ and $\mathrm{P}\left( B \right)=0.45$.  </p>
<p>Find an inequality for&nbsp;$\mathrm{P}\left( A\cap B \right)$.</p>

Events $A$ and $B$ are such that $\mathrm{P}\left( A \right)=0.74$ and $\mathrm{P}\left( B \right)=0.45$.  

Find an inequality for $\mathrm{P}\left( A\cap B \right)$.

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768965663868-ylet0jcwvnl.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5NjU2NjM4NjgteWxldDBqY3d2bmwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwNzkzNiwiZXhwIjoxNzg1NzA4MjM2fQ.PgwWlkKMvV30lMp1l0MSic4uqyhy5a6k_3lv8_NXDSI" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(C'|A')$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(A' \cap C|B)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(A \cup B'|C')$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(C'|B \cap A)$</span></div></li></ol>

<p>Probability of $A$, $\mathrm{P}\left( A \right)=$</p>
<p style="margin-bottom: 1.5rem">Probability of $A'$, $\mathrm{P}\left( A' \right)=$</p>
<p>Probability of $B$, $\mathrm{P}\left( B \right)=$</p>
<p style="margin-bottom: 1.5rem">Probability of $B'$, $\mathrm{P}\left( B' \right)=$</p>
<p style="margin-bottom: 1.5rem">Probability of $A\cap B$, $\mathrm{P}\left( A\cap B \right)=$<br>Probability of $A'\cap B'$, $\mathrm{P}\left( A'\cap B' \right)=$</p>
<p style="margin-bottom: 1.5rem">Probability of $B$ given that&nbsp;$A$ happens, $\mathrm{P}\left( B|A \right)=$<br>Probability of $B'$ given that&nbsp;$A$ happens, $\mathrm{P}\left( B'|A \right)=$<br>Probability of $B$ given that&nbsp;$A'$ happens, $\mathrm{P}\left( B|A' \right)=$<br>Probability of $B'$ given that&nbsp;$A'$ happens, $\mathrm{P}\left( B'|A' \right)=$<br></p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 360px;"><colgroup><col style="width: 120px;"><col style="width: 120px;"><col style="width: 120px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$A$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$A'$</th></tr></thead><tbody><tr style="height: 23px;"><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">$w$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$x$</td></tr><tr style="height: 23px;"><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">$y$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$z$</td></tr></tbody></table></div></div>

Probability of $A$, $\mathrm{P}\left( A \right)=$

Probability of $A'$, $\mathrm{P}\left( A' \right)=$

Probability of $B$, $\mathrm{P}\left( B \right)=$

Probability of $B'$, $\mathrm{P}\left( B' \right)=$

Probability of $A\cap B$, $\mathrm{P}\left( A\cap B \right)=$
Probability of $A'\cap B'$, $\mathrm{P}\left( A'\cap B' \right)=$

Probability of $B$ given that $A$ happens, $\mathrm{P}\left( B|A \right)=$
Probability of $B'$ given that $A$ happens, $\mathrm{P}\left( B'|A \right)=$
Probability of $B$ given that $A'$ happens, $\mathrm{P}\left( B|A' \right)=$
Probability of $B'$ given that $A'$ happens, $\mathrm{P}\left( B'|A' \right)=$


Probability Diagram

<p>For events $A$ and $B$, it is given that&nbsp;$\mathrm{P}(A) = \frac{11}{20}$, $\mathrm{P}(A \cup B) = \frac{4}{5}$ and $\mathrm{P}(B|A) = \frac{4}{11}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(B)$,</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\mathrm{P}(A \cap B')$.</span><span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Determine if the events $A$ and $B'$ are mutually exclusive.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>A third event $C$ is such that&nbsp;$\mathrm{P}(C) = 0.6$ and $B$ and $C$ are independent.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find $\mathrm{P}(B \cap C)$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $\mathrm{P}(A \cap B' \cap C)$.</span><span class="inline-marks">[2]</span></div></li></ol>

For events $A$ and $B$, it is given that $\mathrm{P}(A) = \frac{11}{20}$, $\mathrm{P}(A \cup B) = \frac{4}{5}$ and $\mathrm{P}(B|A) = \frac{4}{11}$.

Determine if the events $A$ and $B'$ are mutually exclusive.

A third event $C$ is such that $\mathrm{P}(C) = 0.6$ and $B$ and $C$ are independent.

Find the range of values of $\mathrm{P}(A \cap B' \cap C)$.

2025 MI P2 Q8

<p>The events $A$ and $B$ are such that&nbsp;$\mathrm{P}(B) = \frac{1}{3}$, $\mathrm{P}(A|B) = \frac{2}{5}$ and $\mathrm{P}(A \cup B) = \frac{3}{5}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find $\mathrm{P}(A)$ and hence, determine whether events $A$ and $B$ are independent.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the exact value of $\mathrm{P}(A' \cap B)$.<span class="inline-marks">[1]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$\mathrm{P}(C) = \frac{1}{2}$ and that&nbsp;$A$ and $C$ are mutually exclusive.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find exactly the greatest and least possible values of $\mathrm{P}(B' \cap C)$.<span class="inline-marks">[4]</span></div></li></ol>

The events $A$ and $B$ are such that $\mathrm{P}(B) = \frac{1}{3}$, $\mathrm{P}(A|B) = \frac{2}{5}$ and $\mathrm{P}(A \cup B) = \frac{3}{5}$.

For a third event $C$, it is given that $\mathrm{P}(C) = \frac{1}{2}$ and that $A$ and $C$ are mutually exclusive.

Math Question Bank