Evaluate these binomial probabilities
a. C82 (.2)2 (.8)6
b. C81 (.2)1 (.8)7
c. C80 (.2)0 (.8)8
d. P(X ≤ 1), when n = 8, p = 0.2
To evaluate these binomial probabilities, we can use the formula for binomial probability:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes in n trials,
- C(n, k) is the binomial coefficient, also known as "n choose k," which represents the number of ways to choose k successes from n trials,
- p is the probability of success in a single trial,
- (1-p) is the probability of failure in a single trial,
- k is the number of successes, and
- n is the number of trials.
For each part of the question, let's plug in the given values into the formula:
a. C82 (.2)^2 (.8)^6
= 28 (.04) (.262144)
≈ 0.29127
b. C81 (.2)^1 (.8)^7
= 8 (.2) (.2097152)
≈ 0.03371
c. C80 (.2)^0 (.8)^8
= 1 (1) (.16777216)
≈ 0.16777
d. To find P(X ≤ 1), we need to calculate the probability of having either 0 or 1 success in 8 trials. We can find this by summing the probabilities of P(X = 0) and P(X = 1).
P(X = 0) = C(8, 0) * (0.2)^0 * (1-0.2)^(8-0)
= 1 * 1 * 0.43046721
≈ 0.43047
P(X = 1) = C(8, 1) * (0.2)^1 * (1-0.2)^(8-1)
= 8 * 0.2 * 0.262144
≈ 0.20972
P(X ≤ 1) = P(X = 0) + P(X = 1)
≈ 0.43047 + 0.20972
≈ 0.64019
Therefore, the binomial probabilities are:
a. ≈ 0.29127
b. ≈ 0.03371
c. ≈ 0.16777
d. ≈ 0.64019