Explanation:
To find P(x), the probability of getting exactly x successes in n trials, we can use the binomial probability formula:
P(x) = (nCx) * p^x * (1 - p)^(n - x)
where n is the number of trials, x is the number of successes, p is the probability of success on a given trial, and (nCx) represents the number of combinations of n items taken x at a time.
Given the values n = 16, x = 3, and p = 1/5, we can substitute these values into the formula:
P(3) = (16C3) * (1/5)^3 * (1 - 1/5)^(16 - 3)
To calculate (16C3), we can use the combination formula:
(16C3) = 16! / (3! * (16 - 3)!)
Calculating the factorials:
16! = 16 * 15 * 14 * ... * 3 * 2 * 1
3! = 3 * 2 * 1
(16 - 3)! = 13! = 13 * 12 * ... * 3 * 2 * 1
Substituting the values:
(16C3) = (16 * 15 * 14) / (3 * 2 * 1) = 560
Now, let's calculate P(3):
P(3) = 560 * (1/5)^3 * (4/5)^(16 - 3)
P(3) = 560 * (1/125) * (256/625)
P(3) = 560 * 256 / (125 * 625)
P(3) = 143360 / 78125
P(3) ≈ 0.1835
Therefore, the probability of getting exactly 3 successes in 16 trials, where the probability of success on a given trial is 1/5, is approximately 0.1835.