Question

Blood Type B is relatively rare in the population, with a probability of .10. These problems will be based on randomly selecting three people and determining if they are Blood Type B or Not. It would be good to determine the sample space and convert that sample space to a discrete random variable of the number of people with Blood Type B (outcomes are 0, 1, 2, or 3). a. What is the expected value of the number of people with Blood Type B from a random sample of 3 people. Use 3 decimal places and use the proper rules of rounding. b. What is the variance of the number of people with Blood Type B from a random sample of 3 people. Use 3 decimal places and use the proper rules of rounding. c. What is the standard deviation of the number of people with Blood Type B from a random sample of 3 people. Use 3 decimal places and use the proper rules of rounding.

Answer 1

The expected value of the number of people with Blood Type B in a random sample of 3 people is approximately 0.243, the variance is approximately 0.196, and the standard deviation is approximately 0.443.

Explanation:

To calculate the expected value (mean), variance, and standard deviation of the number of people with Blood Type B in a random sample of 3 people, we can use probability theory.

a. Expected Value (Mean):

The expected value (μ) of a random variable X is given by:

μ = Σ [x * P(x)]

Where x represents the possible values of the random variable, and P(x) is the probability associated with each value. In this case, x can take values 0, 1, 2, or 3 (the number of people with Blood Type B), and the probabilities are as follows:

P(X = 0) = Probability that none of the 3 people have Blood Type B = (0.90)^3

P(X = 1) = Probability that exactly 1 of the 3 people has Blood Type B = 3C1 * (0.10) * (0.90)^2

P(X = 2) = Probability that exactly 2 of the 3 people have Blood Type B = 3C2 * (0.10)^2 * (0.90)

P(X = 3) = Probability that all 3 people have Blood Type B = (0.10)^3

Now, calculate the expected value:

μ = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2) + 3 * P(X = 3)

μ = 0 * (0.90)^3 + 1 * [3C1 * (0.10) * (0.90)^2] + 2 * [3C2 * (0.10)^2 * (0.90)] + 3 * (0.10)^3

μ ≈ 0.243 (rounded to 3 decimal places)

b. Variance:

The variance (σ^2) of a random variable X is given by:

σ^2 = Σ [(x - μ)^2 * P(x)]

Where μ is the mean we calculated in part (a). Calculate the variance for each possible value of X and then sum them up:

σ^2 = (0 - 0.243)^2 * P(X = 0) + (1 - 0.243)^2 * P(X = 1) + (2 - 0.243)^2 * P(X = 2) + (3 - 0.243)^2 * P(X = 3)

σ^2 ≈ 0.196 (rounded to 3 decimal places)

c. Standard Deviation:

The standard deviation (σ) is the square root of the variance:

σ = √σ^2

σ ≈ √0.196 ≈ 0.443 (rounded to 3 decimal places)

So, the expected value of the number of people with Blood Type B in a random sample of 3 people is approximately 0.243, the variance is approximately 0.196, and the standard deviation is approximately 0.443.