Question

If x is a binomial random variable, compute p(x) for each of the following cases:

(a) n=3,x=2,p=0.4 p(x)= ____
(b) n=5,x=4,p=0.9 p(x)= ____
(c) n=4,x=2,p=0.8 p(x)= ____
(d) n=5,x=1,p=0.1 p(x)= ____

Answer 1

The probabilities p(x) for a binomial random variable with given parameters are calculated using the binomial probability formula and correspond to 0.288, 0.32805, 0.1536, and 0.32805 for the respective cases given.

Step-by-step explanation:

To compute p(x) for a binomial random variable X with the given parameters, we can use the binomial probability formula:

The probability of getting exactly x successes in n trials is given by:

P(X = x) = (n choose x) * p^x * q^(n-x)

Where:

• (n choose x) is the binomial coefficient (nCx) which calculates the number of different combinations of x successes can occur in n trials
• p is the probability of success on a single trial
• q is the probability of failure on a single trial (q = 1 - p)

Using the formula, we calculate:

1. For n=3, x=2, and p=0.4, q is 0.6. Therefore, P(X = 2) = (3 choose 2) * (0.4)^2 * (0.6)^1 = 3 * 0.16 * 0.6 = 0.288.
2. For n=5, x=4, and p=0.9, q is 0.1. Therefore, P(X = 4) = (5 choose 4) * (0.9)^4 * (0.1)^1 = 5 * 0.6561 * 0.1 = 0.32805.
3. For n=4, x=2, and p=0.8, q is 0.2. Therefore, P(X = 2) = (4 choose 2) * (0.8)^2 * (0.2)^2 = 6 * 0.64 * 0.04 = 0.1536.
4. For n=5, x=1, and p=0.1, q is 0.9. Therefore, P(X = 1) = (5 choose 1) * (0.1)^1 * (0.9)^4 = 5 * 0.1 * 0.6561 = 0.32805.

These probabilities are based on the binomial distribution which is appropriate for scenarios with a fixed number of independent trials, each with the same probability of success, and only two possible outcomes: success or failure.