Probability

Let A and B be two independent events such that P(A) = ${1 \over 3}$ and P(B) = ${1 \over 6}$. Then, which of the following is TRUE?

A dice is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is :

When a missile is fired from a ship, the probability that it is intercepted is ${1 \over 3}$ and the probability that the missile hits the target, given that it is not intercepted, is ${3 \over 4}$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is :

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = ${5 \over 9}$, is :

A fair die is tossed until six is obtained on it. Let x be the number of required tosses, then the conditional probability P(x $\ge$ 5 | x > 2) is :

When a certain biased die is rolled, a particular face occurs with probability ${1 \over 6} - x$ and its opposite face occurs with probability ${1 \over 6} + x$. All other faces occur with probability ${1 \over 6}$. Note that opposite faces sum to 7 in any die. If 0 < x < ${1 \over 6}$, and the probability of obtaining total sum = 7, when such a die is rolled twice, is ${13 \over 96}$, then the value of x is :

Let E₁ and E₂ be two events such that the conditional probabilities $P({E_1}|{E_2}) = {1 \over 2}$, $P({E_2}|{E_1}) = {3 \over 4}$ and $P({E_1} \cap {E_2}) = {1 \over 8}$. Then :

If $A$ and $B$ are two events such that $P(A)=\frac{1}{3}, P(B)=\frac{1}{5}$ and $P(A \cup B)=\frac{1}{2}$, then $P\left(A \mid B^{\prime}\right)+P\left(B \mid A^{\prime}\right)$ is equal to :

Out of $60 \%$ female and $40 \%$ male candidates appearing in an exam, $60 \%$ candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is :