Final answer:
To calculate the probability of obtaining exactly 14 successes in a binomial distribution with n=16 trials and a success probability of p=0.8, the formula P(X=x) is used with the combination function, the powers of p and q. The calculated probability P(X=14) is 0.2949, after rounding to four decimal places.
Step-by-step explanation:
The question is asking to find the probability P(X = 14) when a random variable X follows a binomial distribution with n = 16 and p = 0.8. To calculate this probability, we use the formula for the binomial probability:
P(X = x) = C(n, x) * p^x * (1-p)^(n-x)
where:
- C(n, x) is the combination of n items taken x at a time
- p is the probability of success on each trial
- 1-p is the probability of failure, or q
- n is the number of trials
- x is the number of successes (which is 14 in this case)
To solve for P(X = 14), we will plug in the values:
P(X = 14) = C(16, 14) * 0.8^14 * (1-0.8)^(16-14)
This results in P(X = 14) being calculated as:
P(X = 14) = 120 * 0.8^14 * 0.2^2
After performing the calculations:
P(X = 14) = 0.2949 (rounded to four decimal places)