Question

A court is conducting a jury selection. Let X be the number of prospective jurors who will be examined until one is admitted as a juror for a trial. Suppose that X is a geometric random variable, and p, the probability of a juror being admitted, is 0.50.

A). Find the mean and the standard deviation of X.
B). Find the probability that more than two prospective jurors must be examined before one is admitted to the jury.

Answer 1

The mean of X is 2 and the standard deviation is approximately 1.414. The probability that more than two prospective jurors must be examined before one is admitted to the jury is 0.25.

Step-by-step explanation:

To find the mean and standard deviation of the geometric random variable X, we can use the formulas:

Mean (μ) = 1/p

Standard Deviation (σ) = √(1-p)/p^2

Given that p = 0.50, we can substitute the value into the formulas:

Mean (μ) = 1/0.50 = 2

Standard Deviation (σ) = √(1-0.50)/0.50^2 = √0.50/0.25 = √2 = 1.414

Therefore, the mean of X is 2 and the standard deviation is approximately 1.414.

To find the probability that more than two prospective jurors must be examined before one is admitted, we need to find the sum of the probabilities of examining two or fewer jurors and subtract it from 1.

Probability (X > 2) = 1 - (Probability (X = 1) + Probability (X = 2))

Probability (X > 2) = 1 - (0.50 + 0.50 * 0.50) = 1 - (0.50 + 0.25) = 1 - 0.75 = 0.25

Therefore, the probability that more than two prospective jurors must be examined before one is admitted to the jury is 0.25.