the local weatherman has been accurate (successful) in his forecast 80% of the time. find probability that he is accurate exactly 3 of the next 5 days.
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To find the probability that the local weatherman is accurate exactly 3 out of the next 5 days, we can use the binomial probability formula. The formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of desired successes
p is the probability of success in a single trial
In this case, the local weatherman has a success rate of 80% or 0.8, so p = 0.8. The number of trials is 5, and we want to calculate the probability of exactly 3 successes.
Applying the formula, we get:
P(3) = (5C3) * 0.8^3 * (1-0.8)^(5-3)
Calculating the combination (5C3), also known as the binomial coefficient, is done by using the formula:
(nCx) = n! / (x! * (n-x)!)
So for this case, (5C3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = 5 * 4 / 2 = 10.
Substituting the values into the binomial probability formula, we have:
P(3) = 10 * 0.8^3 * (1-0.8)^(5-3)
P(3) ≈ 10 * 0.512 * 0.036
P(3) ≈ 0.18432
Therefore, the probability that the local weatherman is accurate exactly 3 out of the next 5 days is approximately 0.18432 or 18.432%.