Probability Distributions

The joint probability of events A and B is 32 percent with the probability of event A being 60 percent and the probability of event B being 50 percent. Based on this information, the conditional probability of event A given event B has occurred is closest to:

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

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

A program consists of two modules executed sequentially. Let $f_1(t)$ and $f_2(t)$ 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

We are given a set X = {x₁ ..............x_n } where x_i = 2ⁱ A sample S $\subseteq $ X is drawn by selecting each x_i independently with probability p_i = 1/2. The expected value of the smallest number in sample S is:

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

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?

There are n stations in a slotted LAN. Each station attempts to transmit with a probability p in each time slot. What is the probability that ONLY one station transmits in a given time slot?

Suppose the letters a, b, c, d, e, f have probabilities $ \dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\dfrac{1}{32},\dfrac{1}{32}$ respectively.

What is the average length of the correct answer to Q.?

Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty?

Consider a random variable X that takes values + 1 and -1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = -1 and +1 are

Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson's distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

Out of all the 2-digit integers between 1 and 100, a number has to be selected at random. What is the probability that the selected number is not divisible by 7?