Q. Compute $100\times P\left(A|B\right)$, if $P\left(B\right)=0.5$ and $P(A\cap B)=0.32$.

Q. Given that $E$ and $F$ are events such that $P\left(E\right)=0.6,P\left(F\right)=0.3$ and $P(E\cap F)=0.2$, find $6P\left(F|E\right)$.

Q. Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ' the sum of numbers on the dice is $4$'.

Q. Mother, father and son line up at random for a family picture
$E:$ son on one end,

Q. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl ?

1. $0.33,0.34$
2. $0.88,0.98$
3. $0.5,0.33$
4. $0.78,0.67$

Q. A die is thrown three times,
$E:4$ appears on the third toss, $F:6$ and $5$ appears respectively on first two tosses.
Determine $36P$ where $P\left(E|F\right)$

Q. Evaluate $P(A\cup B)$, if $2P\left(A\right)=P\left(B\right)=\frac{5}{13}$ and $P\left(A|B\right)=\frac{2}{5}$. Find $P(A\cup B)\times 100$

Q. A black and a red dice are rolled.
(a) Find the conditional probability of obtaining a sum greater than $9$, given that the black die resulted in a $5$.
(b) Find the conditional probability of obtaining the sum $8$, given that the red die resulted in a number less than $4$.

1. $0.33,0.11$
2. $0.51,0.76$
3. $0.56,0.43$
4. $0.11,0.65$

Q. Consider the experiment of throwing a die, if a multiple of $3$ comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail', given that 'at least one die shows a $3$'.

Q. If $P\left(A\right)=\frac{6}{11},P\left(B\right)=\frac{5}{11}$ and $P(A\cup B)=\frac{7}{11}$, find
(i) $P(A\cap B)$
(ii) $P\left(A|B\right)$
(iii) $P\left(B|A\right)$

1. $0.80,0.98,0.87$
2. $0.58,0.58,0.83$
3. $0.42,0.67,0.57$
4. $0.36,0.80,0.66$

Q. A coin is tossed three times, where
(i) $E$ : head on third toss, $F$ : heads on first two tosses
(ii) $E$ : at least tow heads, $F$ : at most two heads
(iii) $E$ : at most two tails, $F$ : at least one tail
Determine $P\left(E|F\right)$

1. $0.50,0.42,0.85$
2. $0.85,0.42,0.30$
3. $.0.42,0.46,0.47$
4. $0.42,0.50,0.85$

Q. If $A$ and $B$ are events such that $P\left(A|B\right)=P\left(B|A\right)$, then

1. $A\subset B$ but $A\ne B$
2. $A=B$
3. $A\cap B=\Phi$
4. $P\left(A\right)=P\left(B\right)$

Q. An instructor has a question bank consisting of $300$ easy True / False questions, $200$ difficult True / False questions, $500$ easy multiple choice questions and $400$ difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question ?

Q. Two coins are tossed once, where
(i) $E$ : tail appears on one coin, $F$ : one coin shows head
(ii) $E$ : no tail appears, $F$ : no head appears
Determine $P\left(E|F\right)$

1. $1,0.2$
2. $0,2$
3. $1,3$
4. $1,0$

Q. If $P\left(A\right)=0.8,P\left(B\right)=0.5$ and $P\left(B|A\right)=0.4$, find
(i) $P(A\cap B)$
(ii) $P\left(A|B\right)$
(iii) $P(A\cup B)$

Q. A fair die is rolled. Consider events $E=\{1,3,5\},F=\{2,3\}$ and $G=\{2,3,4,5\}$ Find
(i) $P\left(E|F\right)$ and $P\left(F|E\right)$
(ii) $P\left(E|G\right)$ and $P\left(G|E\right)$
(iii) $P\left((E\cup F)|G\right)$ and $P\left((E\cap F)|G\right)$

Q. If $P\left(A\right)=\frac{1}{2}$, $P\left(B\right)=0$, then $P\left(A|B\right)$ is

1. $0$
2. $\frac{1}{2}$
3. Not defined
4. $1$

Q. Two cards are drawn at random and without replacement from a pack of $52$ playing cards. Find the probability that both the cards are black.

Q. If $P\left(A\right)=\frac{3}{5}$ and $P\left(B\right)=\frac{1}{5}$, find $100P(A\cap B)$ if $A$ and $B$ are independent events.

Q. A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing $15$ oranges out of which $12$ are good and $3$ are bad ones will be approved for sale.