Final answer:
The probabilities of obtaining a specific number of heads when flipping a coin 100 times with a head probability of 0.20 can be calculated using the binomial distribution formulas. Scenarios given include finding probabilities for exactly 25 heads, more than 30 heads, less than 15 heads, and between 18 and 22 heads. The provided mean (20) and standard deviation (4) are correct, based on the binomial distribution's parameters.
Step-by-step explanation:
To solve the given problems, we need to use the concept of probability. Using the mean and standard deviation provided, we can verify that the mean is 20 and the standard deviation is 4. Now let's solve the problems:
a) The probability of getting exactly 25 heads can be calculated using the binomial probability formula. P(X = k) = (n choose k) * p^k * q^(n-k), where n is the total number of trials, p is the probability of success, q is the probability of failure, and k is the desired number of successes. In this case, n = 100, p = 0.20, q = 0.80, and k = 25. Plugging in the values, we get:
P(X = 25) = (100 choose 25) * 0.20^25 * 0.80^75
b) The probability of getting more than 30 heads can be calculated by adding the probabilities of getting 31, 32, 33,..., 100 heads. P(X > 30) = P(X = 31) + P(X = 32) + ... + P(X = 100). You can calculate each probability using the formula mentioned earlier.
c) The probability of getting less than 15 heads can be calculated by adding the probabilities of getting 0, 1, 2,..., 14 heads. P(X < 15) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 14).
d) The probability of getting between 18 and 22 heads can be calculated by adding the probabilities of getting 18, 19, 20, 21, and 22 heads. P(18 ≤ X ≤ 22) = P(X = 18) + P(X = 19) + P(X = 20) + P(X = 21) + P(X = 22).