Answer:
Explanation:
To find the probability of exactly three successes in eight trials of a binomial experiment with a probability of success of 45%, we can use the binomial probability formula.
The binomial probability formula is given by:
P(k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(k) is the probability of k successes,
C(n, k) is the number of combinations of n items taken k at a time (also known as "n choose k"),
p is the probability of success,
n is the total number of trials.
In this case, k is 3 (number of successes), p is 0.45 (probability of success), and n is 8 (total number of trials).
Plugging these values into the formula, we get:
P(3) = C(8, 3) * 0.45^3 * (1 - 0.45)^(8 - 3)
Using the combination formula, C(8, 3) = 8! / (3! * (8 - 3)!) = 56.
P(3) = 56 * 0.45^3 * 0.55^5
Calculating this, we find:
P(3) ≈ 0.3113
Therefore, the probability of exactly three successes in eight trials with a 45% probability of success is approximately 0.3113 or 31.13%.