 # 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.

##### 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,



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



(ii)

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



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



(b)

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



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



(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.



(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.



##### Suggested Handwritten and Video Solutions      ##### 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 $\text{P}(\,\,A\cup B\,\,)$.



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



(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 $\frac{\text{1}}{\text{15}}$.



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 $\frac{\text{1}}{\text{15}}$.



Find the probability that

(i)

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



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



(ii)

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



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



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 $0.4$.



##### Suggested Handwritten and Video Solutions    Last Part     Last Part ##### 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.



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



(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.



(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.



##### Suggested Handwritten and Video Solutions      ##### 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 $\text{P}\left( A\cup B \right)=\frac{3}{5}$.

Find the exact values of

(i)

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



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



(ii)

$\text{P}\left( A’\cap B \right)$.



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



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 $\text{P}\left( A’\cap C \right)$.



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



(iv)

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



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



##### Suggested Handwritten and Video Solutions          ##### 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 $\text{P}(A\cap B\cap C)$.



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



##### Suggested Handwritten and Video Solutions        