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>For events $A$ and $B$, it is given that&nbsp;$\mathrm{P}(A) = 0.6$, $\mathrm{P}(A \cup B) = 0.7$ and $\mathrm{P}(B \mid A) = 0.3$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{P}(B) = 0.28$ and determine with a reason, if&nbsp;$A$ and $B$ are independent.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">An event $C$ is such that&nbsp;$A$ and $C$ are mutually exclusive and $\mathrm{P}(C) &gt; 0.25$. Find the range of values of $\mathrm{P}(B' \cap C)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $\mathrm{P}(A) = 0.6$, $\mathrm{P}(A \cup B) = 0.7$ and $\mathrm{P}(B \mid A) = 0.3$.

Show that $\mathrm{P}(B) = 0.28$ and determine with a reason, if $A$ and $B$ are independent.

An event $C$ is such that $A$ and $C$ are mutually exclusive and $\mathrm{P}(C) > 0.25$. Find the range of values of $\mathrm{P}(B' \cap C)$.

Probability Bounds and Inequalities

Addition Rule for Overlapping Events

Probability

<p>The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.</p>
<ol class="i-list"><li data-list-item-id="prob-b-item-i" data-list-item-path="prob-b-item-i"><div class="list-item-paragraph">Find $P(A'\cap B')$ and hence prove that the events $A'$ and $B'$ are independent.<span class="inline-marks">[4]</span></div></li></ol>
<p>A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that&nbsp;$B$ and $C$ are independent, while $A$ and $C$ are not independent.</p>
<ol class="i-list" start="2"><li data-list-item-id="prob-b-item-ii" data-list-item-path="prob-b-item-ii"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[5]</span></div></li></ol>

The events $A$ and $B$ satisfy $P(A)=\dfrac{2}{5}$, $P(B')=\dfrac{3}{4}$ and $P(B\mid A)=\dfrac{1}{4}$.

A third event $C$ satisfies $P(C)=\dfrac{1}{2}$ and $P(A\cap B\cap C)=\dfrac{1}{20}$. It is also known that $B$ and $C$ are independent, while $A$ and $C$ are not independent.

Independent Events

Complementary Events

Conditional Probability

Applying Probability Laws

<p>For events $A$ and $B$, it is given that&nbsp;$P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.</p>
<ol class="i-list"><li data-list-item-id="prob-a-item-i" data-list-item-path="prob-a-item-i"><div class="list-item-paragraph">Find $P(B)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-ii" data-list-item-path="prob-a-item-ii"><div class="list-item-paragraph">Determine, with a reason, whether the events $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="prob-a-item-iii" data-list-item-path="prob-a-item-iii"><div class="list-item-paragraph">Find $P(A'\cap C)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="prob-a-item-iv" data-list-item-path="prob-a-item-iv"><div class="list-item-paragraph">Find the range of possible values of $P(A'\cap B'\cap C')$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P(A)=\dfrac{1}{4}$, $P(A\cup B)=\dfrac{5}{8}$ and $P(A\cap B')=\dfrac{1}{8}$.

For a third event $C$, it is given that $P(C)=\dfrac{2}{5}$, $P(A\cap B\cap C)=\dfrac{1}{20}$, and $C$ is independent of $A$.

<p>Two brothers Kai and Leo are duathlon (running and cycling) athletes who train regularly. For each day, the probability that Kai cycles is $\frac{3}{4}$, the probability that he runs is $\frac{3}{5}$ and the probability he does both is $p$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Write down, in terms of $p$, the probability that, on one day, Kai either runs or cycles but not both.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find the range of possible values of $p$.<span class="inline-marks">[2]</span></div></li></ol>
<p>On average, Leo cycles $5$ out of $7$ days in a week. The probability that Leo cycles when Kai cycles is $0.9$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find the probability that Leo cycles when Kai does not.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">State, in context, two assumptions needed for the number of days Leo cycles over a period of $5$ weeks to be well modelled by a binomial distribution.<span class="inline-marks">[2]</span></div></li></ol>
<p>Assume now that the number of days Leo cycles in $5$ weeks has a binomial distribution.</p>
<ol class="abc-list" start="5"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><div class="list-item-paragraph">Find the probability that, in the $5$ weeks, Leo cycles on at least $20$ days but fewer than $30$ days.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-1" data-list-item-path="imported-item-block-5-item-1"><div class="list-item-paragraph">Find the probability that, in the $5$ weeks, there are exactly $5$ days in which both brothers do not cycle and they both cycle on the other days.<span class="inline-marks">[2]</span></div></li></ol>

Two brothers Kai and Leo are duathlon (running and cycling) athletes who train regularly. For each day, the probability that Kai cycles is $\frac{3}{4}$, the probability that he runs is $\frac{3}{5}$ and the probability he does both is $p$.

Write down, in terms of $p$, the probability that, on one day, Kai either runs or cycles but not both.

Find the range of possible values of $p$.

On average, Leo cycles $5$ out of $7$ days in a week. The probability that Leo cycles when Kai cycles is $0.9$.

Find the probability that Leo cycles when Kai does not.

State, in context, two assumptions needed for the number of days Leo cycles over a period of $5$ weeks to be well modelled by a binomial distribution.

Assume now that the number of days Leo cycles in $5$ weeks has a binomial distribution.

Find the probability that, in the $5$ weeks, Leo cycles on at least $20$ days but fewer than $30$ days.

Find the probability that, in the $5$ weeks, there are exactly $5$ days in which both brothers do not cycle and they both cycle on the other days.

2024 RI P2 Q11

Discrete Random Variables (DRV)

<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>Events $A$ and $B$ are such that&nbsp;$\mathrm{P}(A)=0.74$ and $\mathrm{P}(B)=0.45$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $\mathrm{P}(A \cap B)$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $\mathrm{P}(A \cup B)$.</span></div></li></ol>

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

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

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

IP2U10 HOT Q3

<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 $\mathrm{P}(A \cup C \cup D)$.

Finding Unknowns in Probability Models

2025 VJC P2 Q6

<p>A code consists of $10$ characters.</p>
<p>The first $5$ characters of the code is formed using $5$ letters chosen from the set&nbsp;$\{A, B, C, D, E, F, G, H\}$ and the last $5$ characters of the code is formed using $5$ digits chosen from the set&nbsp;$\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0\}$. The code allows for repetitions of letters and/or digits.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of different codes that can be formed.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability that a code chosen at random</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">contains the letter $E$ exactly thrice and the number $5$ exactly once,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">has&nbsp;$E$ as the first character or $4$ as its tenth character, but not both.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

The first $5$ characters of the code is formed using $5$ letters chosen from the set $\{A, B, C, D, E, F, G, H\}$ and the last $5$ characters of the code is formed using $5$ digits chosen from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0\}$. The code allows for repetitions of letters and/or digits.

Find the number of different codes that can be formed.

Find the probability that a code chosen at random

Arrangements with Repetition Allowed

Arrangements with Position or Pattern Restrictions

Counting with Inclusion-Exclusion

2025 VJC P2 Q5

Permutation and Combination

<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.

Probabilities of Successive Events

Probability Using Permutations and Combinations

Waiting-Time and Stopping-Rule Probabilities

2013 HCI P2 Q10

<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.

2025 JPJC P2 Q9

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\mathrm{P}(B|A) = 0.2$, $\mathrm{P}(A|B) = 0.6$ and $\mathrm{P}(A' \cap B') = 0.3$, find $\mathrm{P}(A \cap B)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The events $X$, $Y$ and $Z$ are such that the events $X$ and $Z$ are mutually exclusive. Given further that&nbsp;$\mathrm{P}(X \cap Y) = 0.1$, $\mathrm{P}(X' \cap Y) = 0.35$, $\mathrm{P}(Y \cap Z) = 0.2$, and $\mathrm{P}(X \cup Y \cup Z) = 0.95$, find the minimum and maximum values of $\mathrm{P}(X)$.</span><span class="inline-marks">[3]</span></div></li></ol>

If $\mathrm{P}(B|A) = 0.2$, $\mathrm{P}(A|B) = 0.6$ and $\mathrm{P}(A' \cap B') = 0.3$, find $\mathrm{P}(A \cap B)$.

The events $X$, $Y$ and $Z$ are such that the events $X$ and $Z$ are mutually exclusive. Given further that $\mathrm{P}(X \cap Y) = 0.1$, $\mathrm{P}(X' \cap Y) = 0.35$, $\mathrm{P}(Y \cap Z) = 0.2$, and $\mathrm{P}(X \cup Y \cup Z) = 0.95$, find the minimum and maximum values of $\mathrm{P}(X)$.

2025 ACJC P2 Q6

<p>For events $A$ and $B$, it is given that&nbsp;$\mathrm{P}\left( B \right)=\frac{7}{27}$, $\mathrm{P}\left( B|A \right)=\frac{2}{9}$ and $\mathrm{P}\left( A|B' \right)=\frac{7}{10}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with reason, whether $A$ and $B$are independent.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cup B \right)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find $\mathrm{P}\left( A\cap B \right)$.<span class="inline-marks">[3]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$\mathrm{P}\left( B\cap C \right)=\frac{1}{18}$ and that&nbsp;$A$ and $C$ are mutually exclusive.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the range of $\mathrm{P}\left( C \right)$.<span class="inline-marks">[2]</span></div></li></ol>

For events $A$ and $B$, it is given that $\mathrm{P}\left( B \right)=\frac{7}{27}$, $\mathrm{P}\left( B|A \right)=\frac{2}{9}$ and $\mathrm{P}\left( A|B' \right)=\frac{7}{10}$.

For a third event $C$, it is given that $\mathrm{P}\left( B\cap C \right)=\frac{1}{18}$ and that $A$ and $C$ are mutually exclusive.

2015 PJC P2 Q11

<p>There are $47$ cards in a deck of which $22$ are picture cards and $25$ are number cards. Each card has a suit of either a clubs or a heart. There is a total of $27$ cards with the suit of clubs, $18$ of which are number cards. </p>
<p>A card is chosen at random from the deck, find the probability of choosing</p>
<ol class="i-list"><li><div class="list-item-paragraph">a picture card,</div></li><li><div class="list-item-paragraph">a number card with a suit of clubs,</div></li><li><div class="list-item-paragraph">a picture card with a suit of heart,</div></li><li><div class="list-item-paragraph">a picture card or a card with a suit of heart.</div></li></ol>

There are $47$ cards in a deck of which $22$ are picture cards and $25$ are number cards. Each card has a suit of either a clubs or a heart. There is a total of $27$ cards with the suit of clubs, $18$ of which are number cards. 

A card is chosen at random from the deck, find the probability of choosing

Probability of a Single Event

Basic Probability: Picture and Number Cards

<p>A bag contains $24$ candies of which half of them are orange flavour and the remaining are apple flavour. The candies are wrapped in either black or white paper. There are $8$ candies wrapped in white paper, $5$ of which are apple flavour. </p>
<p>A candy is chosen at random from the bag, find the probability of choosing </p>
<ol class="i-list"><li><div class="list-item-paragraph">a candy with apple flavour,</div></li><li><div class="list-item-paragraph">a candy with orange flavour wrapped in black paper,</div></li><li><div class="list-item-paragraph">a candy with orange flavour or a candy wrapped in black paper. </div></li></ol>

A bag contains $24$ candies of which half of them are orange flavour and the remaining are apple flavour. The candies are wrapped in either black or white paper. There are $8$ candies wrapped in white paper, $5$ of which are apple flavour. 

A candy is chosen at random from the bag, find the probability of choosing 

Probability: Choosing a Candy from a Mixed Collection

<p>There are 35 cards in a deck of which 8 are picture cards and 27 are number cards. Each card has a suit of either a spade or a heart. There is a total of 19 cards with the suit of spade, 12 of which are number cards. </p>
<p>A card is chosen at random from the deck, find the probability of choosing</p>
<ol class="i-list"><li><div class="list-item-paragraph">a picture card,</div></li><li><div class="list-item-paragraph">a number card with a suit of spade,</div></li><li><div class="list-item-paragraph">a picture card with a suit of heart,</div></li><li><div class="list-item-paragraph">a picture card or a card with a suit of heart.</div></li></ol>

There are 35 cards in a deck of which 8 are picture cards and 27 are number cards. Each card has a suit of either a spade or a heart. There is a total of 19 cards with the suit of spade, 12 of which are number cards. 

Probability with a Modified Deck of Cards

<p>Jane has 25 books on her desk of which 17 are novels and 8 are comic books. The books are either in Spanish or English medium. She has a total of 10 Spanish books, half of which are comic books. </p>
<p>If each book is equally likely to be chosen and a book is chosen at random from the desk, find the probability of choosing</p>
<ol class="i-list"><li data-list-item-id="item-1765277037552-vtu1ye77j" data-list-item-path="item-1765277037552-vtu1ye77j"><div class="list-item-paragraph">an English book,</div></li><li data-list-item-id="item-1765277039089-j5oqeddxf" data-list-item-path="item-1765277039089-j5oqeddxf"><div class="list-item-paragraph">a Spanish novel,</div></li><li data-list-item-id="item-1765277039524-hbtkvdakz" data-list-item-path="item-1765277039524-hbtkvdakz"><div class="list-item-paragraph">a novel,</div></li><li data-list-item-id="item-1765277040019-bhn2ra9qu" data-list-item-path="item-1765277040019-bhn2ra9qu"><div class="list-item-paragraph">an English comic book,</div></li><li data-list-item-id="item-1765277041242-ex3oy5br0" data-list-item-path="item-1765277041242-ex3oy5br0"><div class="list-item-paragraph">an English or novel book.</div></li></ol>

Jane has 25 books on her desk of which 17 are novels and 8 are comic books. The books are either in Spanish or English medium. She has a total of 10 Spanish books, half of which are comic books. 

If each book is equally likely to be chosen and a book is chosen at random from the desk, find the probability of choosing

Probability: Choosing a Book from a Mixed Collection

<p>Dylan has 17 books on his desk of which 12 are textbooks and 5 are comic books. The books are either in Japanese or English medium. He has a total of 6 Japanese books, 4 of which are comic books. </p>
<p>If each book is equally likely to be chosen and a book is chosen at random from the desk, find the probability of choosing</p>
<ol class="i-list"><li><div class="list-item-paragraph">a book in English,</div></li><li><div class="list-item-paragraph">a Japanese textbook,</div></li><li><div class="list-item-paragraph">a comic book,</div></li><li><div class="list-item-paragraph">an English comic book,</div></li><li><div class="list-item-paragraph">an English or comic book. </div></li></ol>

Dylan has 17 books on his desk of which 12 are textbooks and 5 are comic books. The books are either in Japanese or English medium. He has a total of 6 Japanese books, 4 of which are comic books. 

Basic Probability: Categorising Books

<p>In a survey of $140$ students who had recently migrated with their families to Singapore, students were asked to indicate the birth places of both their parents. The results are displayed in the accompanying table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><tbody><tr style="height: 30px;"><td colspan="2" rowspan="2" style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"></td><td colspan="3" style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">Mother’s birth place</td></tr><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">UK</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">Europe</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">Asia</td></tr><tr style="height: 30px;"><td rowspan="3" style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>Father’s birth place</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>UK</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$45$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$16$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$4$</td></tr><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>Europe</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$7$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$12$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$1$</td></tr><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center"><p>Asia</p></td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$8$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$2$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$45$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1760783275343-g48xz2e43" data-list-item-path="item-1760783275343-g48xz2e43"><div class="list-item-paragraph">Find the probability that a randomly selected student from this group has both parents born in the UK.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760783275831-gsbc0aj4s" data-list-item-path="item-1760783275831-gsbc0aj4s"><div class="list-item-paragraph">Find the probability that a student randomly selected from those with mothers born in the UK, has a father born in Asia.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760783276355-paukr8n5b" data-list-item-path="item-1760783276355-paukr8n5b"><div class="list-item-paragraph">Find the probability that a student randomly chosen from this group have a parent born in either UK or Europe.<span class="inline-marks">[3]</span></div></li></ol>

In a survey of $140$ students who had recently migrated with their families to Singapore, students were asked to indicate the birth places of both their parents. The results are displayed in the accompanying table.

Conditional Probability from Tables

2015 NYJC P2 Q8

<p>For events $A$ and $B$, it is given that&nbsp;$P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760781172283-qq04lwcra" data-list-item-path="item-1760781172283-qq04lwcra"><div class="list-item-paragraph">Find $P\left( B \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781174061-kzrl5fl7k" data-list-item-path="item-1760781174061-kzrl5fl7k"><div class="list-item-paragraph">Determine, with a reason, whether the events $A$ and $B$ are independent.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a third event $C$, it is given that&nbsp;$P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760781194927-07nr12r5q" data-list-item-path="item-1760781194927-07nr12r5q"><div class="list-item-paragraph">Find $P\left( {A}'\cap C \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760781195819-twm664oux" data-list-item-path="item-1760781195819-twm664oux"><div class="list-item-paragraph">Find the range of values for&nbsp;$P\left( A'\cap {B}'\cap {C}' \right)$.<span class="inline-marks">[3]</span></div></li></ol>

For events $A$ and $B$, it is given that $P\left( A \right)=\frac{1}{8}$, $P\left( A\cup B \right)=\frac{9}{16}$ and $P\left( A\cap {B}' \right)=\frac{1}{16}$.

For a third event $C$, it is given that $P\left( C \right)=\frac{3}{8}$, $P\left( A\cap B\cap C \right)=\frac{1}{64}$ and $C$ is independent of $A$.

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