# Probability

Probability is the measure of the likelihood of occurrence of an event. It is a numerical value that expresses the chance that a given event will occur. It has a range from $0$ to $1$, including both limits. It can be written as a fraction, decimal or percentage. A probability of $0$ means that there is no chance of an event occurring. A probability of $1$ means that an event is certain to occur.Â

##### 2012 SAJC P2 Q7

(a)

Given that the two events $A$ and $B$ are such that $\text{P}(A|B)=\frac{2}{3},\,\,\text{P}(A\cap B’\,)=\frac{1}{4}$ and $\text{P}(A\cap B)=\frac{5}{12}$,

(a)Given that the two events $A$ and $B$ are such that $\text{P}(A|B)=\frac{2}{3},\,\,\text{P}(A\cap B’\,)=\frac{1}{4}$ and $\text{P}(A\cap B)=\frac{5}{12}$,

(i)

determine if $A$ and $B$ are independent,

[1]

(i) determine if $A$ and $B$ are independent,

[1]

(ii)

find $\text{P}(A\cup B)$.

[2]

(ii) find $\text{P}(A\cup B)$.

[2]

(b)

Ali and Bob take turns to shoot an arrow in an Archery training session. The probability that Ali hits the bullâ€™s eye is $0.3$ and for Bob, it is $0.4$. Ali shoots first. Find the probability that Ali hits the bullâ€™s eye first.

[3]

(b) Ali and Bob take turns to shoot an arrow in an Archery training session. The probability that Ali hits the bullâ€™s eye is $0.3$ and for Bob, it is $0.4$. Ali shoots first.
Find the probability that Ali hits the bullâ€™s eye first.

[3]

(c)

Calvin, his parents and $7$ other people are seated in a round table of $10$. Find the probability that his parents are seated together but Calvin is not seated beside either of his parents.

[2]

(c) Calvin, his parents and $7$ other people are seated in a round table of $10$. Find the probability that his parents are seated together but Calvin is not seated beside either of his parents.

[2]

##### 2012 RVHS P2 Q10

(a)

For events $A$ and $B$, it is given that $\text{P}(B)=0.4$ and $\text{P}(\,\,A|B’\,\,)=0.15$. Find $\text{P}(\,\,A\cup B\,\,)$.

[3]

(a) For events $A$ and $B$, it is given that $\text{P}(B)=0.4$ and $\text{P}(\,\,A|B’\,\,)=0.15$. Find $\text{P}(\,\,A\cup B\,\,)$.

[3]

(b)

In the last stage of a variety game show, the contestant aims to open a treasure chest containing the grand prize. The chest can be opened only when its two locks are unlocked. Each lock requires a unique key to be unlocked. The contestant is given up to three rounds to select the two correct keys among six different keys on a tray. For each round, he can only select one key which will be tried on both locks. The selected key of a particular round will be removed from the tray after that round. Show that the probability that a contestant wins the grand prize in the second round is $\frac{\text{1}}{\text{15}}$.

[1]

Find the probability that
In the last stage of a variety game show, the contestant aims to open a treasure chest containing the grand prize. The chest can be opened only when its two locks are unlocked. Each lock requires a unique key to be unlocked. The contestant is given up to three rounds to select the two correct keys among six different keys on a tray. For each round, he can only select one key which will be tried on both locks. The selected key of a particular round will be removed from the tray after that round. Show that the probability that a contestant wins the grand prize in the second round is $\frac{\text{1}}{\text{15}}$.

[1]

Find the probability that

(i)

the grand prize is won and the contestant selects the correct key in the first round,

[2]

(i) the grand prize is won and the contestant selects the correct key in the first round,

[2]

(ii)

the contestant selects the correct key in the first round given that he wins the grand prize.

[2]

(ii) the contestant selects the correct key in the first round given that he wins the grand prize.

[2]

The variety game show runs for five days a week, over a period of $60$ weeks. Find the probability that the mean number of shows per week for which the grand prize is won in the second round is more than $0.4$.

[3]

Last Part

Last Part

Â

##### 2018 SRJC P2 Q10 (b)

For events $A$ and $B$, it is given that $\text{P}\left( A \right)=\frac{11}{20}$, $\text{P}\left( B \right)=\frac{2}{5}$ and $\text{P}\left( A\cap B \right)=\frac{1}{4}$.

(i)

Find the probability where event $A$ occurs or event $B$ occurs but not both.

[1]

(i) Find the probability where event $A$ occurs or event $B$ occurs but not both.

[1]

(ii)

show that ${{u}_{3}}+{{u}_{4}}+{{u}_{5}}+…+{{u}_{n}}=\,p\left[ 1-{{\left( -\frac{1}{3} \right)}^{n+q}} \right]$, where the integers $p$ and $q$ are to be found.

[3]

(ii) show that ${{u}_{3}}+{{u}_{4}}+{{u}_{5}}+…+{{u}_{n}}=\,p\left[ 1-{{\left( -\frac{1}{3} \right)}^{n+q}} \right]$, where the integers $p$ and $q$ are to be found.

[3]

##### 2019 JPJC P2 Q11

The events $A$ and $B$ are such that $\text{P}\left( A|B \right)=\frac{7}{10}$, $\text{P}\left( B|A \right)=\frac{4}{15}$ and $\text{P}\left( A\cup B \right)=\frac{3}{5}$.Â

Find the exact values of

(i)

$\text{P}\left( A\cap B \right)$,

[3]

(i) $\text{P}\left( A\cap B \right)$,

[3]

(ii)

$\text{P}\left( A’\cap B \right)$.

[2]

(ii) $\text{P}\left( A’\cap B \right)$.

[2]

Find a third event $C$, it is given that $\text{P}\left( C \right)=\frac{3}{10}$ and that $A$ and $C$ are independent.

(iii)

Find $\text{P}\left( A’\cap C \right)$.

[2]

(iii) Find $\text{P}\left( A’\cap C \right)$.

[2]

(iv)

Hence state an inequality satisfied by $\text{P}\left( A’\cap B\cap C \right)$.

[1]

(iv) Hence state an inequality satisfied by $\text{P}\left( A’\cap B\cap C \right)$.

[1]

##### 2020 ACJC P2 Q5

For events $A$ and $B$, it is given that $P(A\cup B)=0.8$ and $P(\left. A \right|B)=0.5$. It is also given that events $A$ and $B$ are independent.

(i)

Find $\text{P(}B\text{)}$.

(i) Find $\text{P(}B\text{)}$.

A third event $C$ is such that events $A$ and $C$ are independent and events $B$ and $C$ are independent.

(ii)

Given also that $\text{P}(C)=0.5$, find exactly the maximum and minimum possible values of $\text{P}(A\cap B\cap C)$.

[3]

Given also that $\text{P}(C)=0.5$, find exactly the maximum and minimum possible values of $\text{P}(A\cap B\cap C)$.

[3]