Question

Suppose that in a certain metropolitan area, nine out of 10 households have cable TV. Let x denote the number among four randomly selected households that have cable TV, so x is a binomial random variable with n = 4 and p = 0.9. (Round your answers to four decimal places.)

Calculate p(2) = P(x = 2).

Answer 1

To find the probability that exactly two out of four households have cable TV when nine out of ten have it, we use the binomial probability formula, with n = 4, k = 2, and p = 0.9, yielding a result of 0.0486.

Step-by-step explanation:

The question deals with the calculation of the probability of getting a specific number of successes in a binomial distribution. For this particular scenario, we are given that x is a binomial random variable with n = 4 trials and a success probability of p = 0.9. The task is to calculate P(x = 2), which means finding the probability that exactly two out of four randomly selected households have cable TV.

To solve this, we use the binomial probability formula:

P(x = k) = C(n, k) * p^k * (1-p)^(n-k)

Where:

• C(n, k) is the combination of n items taken k at a time
• p is the probability of success on a single trial
• k is the number of successes
• n is the total number of trials

Substituting the values we have:

P(x = 2) = C(4, 2) * 0.9^2 * (1-0.9)^(4-2)

Simplifying, we get:

P(x = 2) = 6 * 0.81 * 0.01

P(x = 2) = 0.0486

The probability that exactly two out of the four households selected at random have cable TV is 0.0486, rounded to four decimal places.