Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places.
n = 64, x = 3, p = 0.04
(1 point)
Responses
0.091
0.091
0.139
0.139
0.221
0.221
0.375
To calculate the probability of x successes in a binomial distribution, you can use the binomial probability formula:
P(x) = C(n,x) * p^x * (1-p)^(n-x)
where:
P(x) is the probability of x successes,
C(n,x) is the combination formula, or the number of ways to choose x successes out of n trials,
p is the probability of success on a single trial, and
(1-p) is the probability of failure on a single trial.
Using the given values of n = 64, x = 3, and p = 0.04, we can calculate the probability:
P(3) = C(64,3) * 0.04^3 * (1-0.04)^(64-3)
Using the combination formula C(n,x) = n! / (x!(n-x)!), we can calculate:
C(64,3) = 64! / (3!(64-3)!) = 64! / (3! * 61!) = (64 * 63 * 62) / (3 * 2 * 1) = 41664
Now, we can substitute the known values into the formula:
P(3) = 41664 * 0.04^3 * 0.96^61
Calculating further:
P(3) = 41664 * 0.000064 * 7.371e-05
P(3) ≈ 0.091
Therefore, the probability of 3 successes in 64 trials with a success probability of 0.04 is approximately 0.091, rounded to three decimal places.