## Probability Questions and Answers Part-5

1. A group of 6 boys and 6 girls is randomly divided into two equal groups. The probability that each group contains 3 boys and 3 girls is
a) 10/231
b) 5/231
c) 90/231
d) 100/231

Explanation: The number of ways of choosing 6 persons out 2. In a hurdle race, a runner has probability p of jumping over a specific hurdle . Given that in 5 trials, the runner succeeded 3 times, the conditional probability that the runner had succeeded in the first trial is
a) $\frac{3}{5}$
b) $\frac{2}{5}$
c) $\frac{1}{5}$
d) $\frac{4}{5}$

Explanation: Let A denote the event that the runner succeeds  3. Let F denote the set of all onto functions from $A=\left\{a_{1},a_{2},a_{3},a_{4}\right\}$     to $B=\left\{ x,y,z,\right\}$    . A function f is chosen at random from F. The probability that f–1{x} consists of exactly one element is
a) 2/3
b) 1/3
c) 1/6
d) 0

Explanation: Let us first count the number of elements in F 4. Three integers are chosen at random without replacement from the first 20 integers. The probability that their product is even is
a) 7/19
b) 2/19
c) 17/19
d) 10/19

Explanation: The product of any number of integers will be  5. A box contains tickets numbered 1 to N. n tickets are drawn from the box with replacement. The probability that the largest number on the tickets is k is
a) $\left(\frac{k}{N}\right)^{n}$
b) $\left(\frac{k-1}{N}\right)^{n}$
c) 0
d) $\frac{k^{n}-\left(k-1\right)^{n}}{N^{n}}$

Explanation: Let X denote the largest number on the n tickets drawn.

6. A biased cubical die is thrown twice. Let
P = probability of getting the same number both the times.
Q= probability getting odd number on the first thrw and an even number on the second throw.
Then
a) $Q \leq\frac{1}{4}$
b) $P \geq\frac{1}{3}-\frac{2}{3}Q$
c) $P +2Q\leq 1$
d) All of the Above

Explanation: Let pi = probability of getting number i on the  7. An unbiased cubical die is thrown 2n times, and let P denote the probability of getting even numbers at least n times, then
a) $P >\frac{1}{2}$
b) $P \geq\frac{1}{2}+\frac{1}{4n}$
c) $P =\frac{163}{256}$   if n=4
d) All of the Above

Explanation: Let p = probability of getting an even number on a single throw of the die
and let X = number of times an even number is obtained   8. If A and B are two events, the probability that exactly one of them occurs is given by
a) $P\left(A\right)+P\left(B\right)-2P\left(A\cap B\right)$
b) $P\left(A\cap B'\right)+P\left(A'\cap B\right)$
c) $P\left(A\cup B\right)-P\left(A\cap B\right)$
d) All of the Above

Explanation: We have P(exactly one of A, B occurs) 9. If A and B are two events such that P(A) = 1/2 and P(B) = 2/3, then
a) $P\left(A\cup B\right)\geq 2/3$
b) $P\left(A\cap B'\right)\leq 1/3$
c) $1/16\leq P\left(A\cap B\right)\leq 1/2$
d) All of the Above a) P (exactly two of A, B, C occur) $\leq P\left(A \cap B\right) + P\left(B \cap C\right) + P\left(C \cap A\right)$
b) $P\left(A\cup B \cup C\right) \leq P\left(A\right) + P\left(B\right) + P\left(C\right)$
c) P (exactly one of A, B, C occur) $\leq P\left(A\right) + P\left(B\right) + P\left(C\right) – P\left(B \cap C\right) – P\left(C\cap A\right) – P\left(A \cap B\right)$ 