Probability of X=Y for Independent Binomial Random Variables

Published on May 5, 2026

Statistics

Calculate the probability P(X=Y) where X ~ B(1, 1/3) and Y ~ B(2, 1/2) are independent binomial random variables.

Question

Answer

The problem states that random variables

and

are independent.

follows a binomial distribution

, which means

can take values 0 or 1. The probability mass function (PMF) of

is:

For

, we have

and

follows a binomial distribution

, which means

can take values 0, 1, or 2. The PMF of

is:
For

, we have

and

We want to find

. Since

can only take values 0 and 1, and

can take values 0, 1, and 2, the only possible values for

are when

.
Thus,

Since

and

are independent,

.
So,

Substitute the probabilities we calculated:

The final answer is

Theorems

• Binomial Distribution

The binomial distribution is a discrete probability distribution that expresses the probability of

successes in a fixed number of

independent Bernoulli trials, each with a probability of success

. The probability mass function is given by



• Independence of Random Variables

Two random variables

and

are independent if the probability of their joint occurrence is the product of their individual probabilities. That is,

for all possible values of

and

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