Final answer:
To find the probability of getting exactly three successes in a binomial distribution with n=5 and p=0.2, we use the binomial probability formula. The calculation gives us a probability of 0.0512, which means there is a 5.12% chance of exactly three successes.
Step-by-step explanation:
To determine the probability of exactly three successes in a binomial probability distribution where n=5 and p=0.2, we use the binomial probability formula:
P(x = 3) = C(5, 3) * (0.2)^3 * (0.8)^2
First calculate the combination for selecting 3 successes out of 5 trials: C(5, 3) = 5! / (3! * (5-3)!). Calculating this, we get 10.
Next, raise the probability of success, 0.2, to the third power since we want three successes and raise the probability of failure, 0.8, to the second power since we want two failures (remembering that q = 1 - p):
(0.2)^3 = 0.008 and (0.8)^2 = 0.64.
Multiplying these together along with the combination we just calculated:
10 * 0.008 * 0.64 = 0.0512.
Therefore, the probability of getting exactly three successes is 0.0512 or 5.12%.