Final answer:
The probability of getting exactly 3 successes in 5 trials with a probability of 1/2 of success, calculated using the binomial distribution formula, is 5/16. This is option 2 in the list of possible answers.
Step-by-step explanation:
The student is asking about the probability of getting exactly 3 successes in 5 trials of Bernoulli trials with a success probability of 1/2. This can be calculated using the binomial distribution formula, which is given by P(X = k) = C(n, k) * pk * q(n-k), where:
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of success,
- q is the probability of failure, and
- k is the number of successes.
In this case, we are looking for the probability of getting 3 successes (k = 3) in 5 trials (n = 5), where the probability of success (p) is 1/2, and therefore the probability of failure (q) is also 1/2. Calculating the combinations, C(5, 3) = 10, and using these values in the binomial distribution formula yields:
P(X = 3) = C(5, 3) * (1/2)3 * (1/2)2 = 10 * (1/8) * (1/4) = 10/32 = 5/16.
Therefore, the correct answer is option 2) 5/16.