Question

An urn contains 2 red and 3 blue balls. Balls are drawn from the urn, one after another and without replacement, until a red ball is produced, at which time the experiment ends. The random variable X records the total number of balls remaining in the urn at the end of the experiment. Find E[X]. Enter the exact expected value as a whole number. For example: 1, 3, 5, and 8 are all examples of whole numbers.

Answer 1

The expected value of the random variable X, representing the total number of balls remaining in the urn at the end of the experiment, is 1/2.

Step-by-step explanation:

The expected value, or mean, of a random variable X can be calculated by multiplying each possible value of X by its corresponding probability and summing them all.

In this case, the random variable X represents the total number of balls remaining in the urn at the end of the experiment. The possible values of X are 0, 1, and 2, since there can be 0, 1, or 2 blue balls left in the urn when a red ball is drawn.

Let's calculate the expected value:

1. The probability that the experiment ends with 0 blue balls remaining is the probability of drawing a red ball first, which is 2/5.
2. The probability that the experiment ends with 1 blue ball remaining is the probability of drawing a blue ball first and then a red ball, which is (3/5) * (2/4) = 3/10.
3. The probability that the experiment ends with 2 blue balls remaining is the probability of drawing two blue balls first and then a red ball, which is (3/5) * (2/4) * (1/3) = 1/10.

Now, we can calculate the expected value:

E(X) = 0 * (2/5) + 1 * (3/10) + 2 * (1/10) = 0 + 3/10 + 2/10 = 5/10 = 1/2.

Therefore, the expected value of X is 1/2.