Probability Theory Practice Questions

Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1, 2,3 ,...,20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?

In a multi-user operating system on an average, 20 requests are made to use a particular resource per hour. The arrival of requests follows a Poisson distribution. The probability that either one, three or five requests are made in 45 minutes is given by :

When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that successive tosses are independent, the expected value of N is

For each element in set of size 2n, an unbiased coin is tossed. The 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is

In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data indicates that if the temperature at noon is less than or equal to $25^{\circ}C$ , the probability that it will rain in the afternoon is 0.4. The temperature at noon is equally likely to be above $25^{\circ}C$ , or at/below $25^{\circ}C$ . What is the probability that it will rain in the afternoon on a day when the temperature at noon is above $25^{\circ}C$ ?

A bag contains 10 blue marbles, 20 green marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag. This process is repeated 3 times. The probability that no two of the marbles drawn have the same colour is

In a communication network, a packet of length L bits takes link $L_1$ with a probability of $p_1$ or link $L_2$ with a probability of $p_2$ . Link $L_1$ and $L_2$ have bit error probability of $b_1$ and $b_2$ respectively. The probability that the packet will be received without error via either $L_1$ or $L_2$ is

An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the trials are independent, the expected number of tosses is

Box P has 2 red balls and 3 blue balls and box Q has 3 balls and 1 blue ball. A ball is selected as follows: (i) select a box (ii) choose a ball from the selected box such that each ball in the box is equally likely to be chosen. The probabilities of selecting boxes P and Q are 1/3 and 2/3 respectively. Given that a ball selected in the above process is a red ball, the probability that it came from the box P is:

A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that two such randomly generated strings are not identical is:

Let f(x) be the continuous probability density function of a random variable X. The probability that $a \lt X \leq b$ , is:

In a population of N families, 50% of the families have three children, 30% of the families have two children and the remaining families have one child. What is the probability that a randomly picked child belongs to a family with two children?

Two n bit binary strings, S1 and S2, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

An examination paper has 150 multiple choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 marks. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained by all these students is

Let $X$ and $Y$ be two exponentially distributed and independent random variables with mean $\alpha \text{ and }\beta$ , respectively. If $Z=min(X,Y)$ , then the mean of $Z$ is given by

If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A|B) and P(B|A) respectively are

A program consists of two modules executed sequentially. Let f1(t) and f2(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by