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.
n=21, x=18, p=0.75
To find the probability of x successes in a binomial distribution, we use the binomial probability formula:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where:
- P(x) is the probability of getting x successes
- n is the number of trials
- x is the number of successful outcomes
- p is the probability of success on a single trial
- (nCx) is the combination, also known as the binomial coefficient, which represents the number of ways to choose x successes from n trials.
Given that n = 21, x = 18, and p = 0.75, we can substitute these values into the formula and calculate the probability of getting 18 successes:
P(18) = (21C18) * (0.75^18) * (0.25^3)
To calculate (21C18), we use the combination formula:
(21C18) = 21! / (18! * (21-18)!)
Substituting the values:
(21C18) = 21! / (18! * 3!)
21! = 21 * 20 * 19 * 18!
After simplification:
(21C18) = (21 * 20 * 19) / (3 * 2 * 1)
Now we can substitute all the values into the binomial probability formula:
P(18) = [(21 * 20 * 19) / (3 * 2 * 1)] * (0.75^18) * (0.25^3)
By evaluating this expression, we can find the probability of getting 18 successes in 21 trials with a single trial success probability of 0.75.