if a rifleman averages 8 hits out of 10 shots at a target, what is the probability that he will hit the target in 3 out of 4 shots
To find the probability that a rifleman will hit the target in 3 out of 4 shots, we can use the binomial probability formula.
The binomial probability formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where:
P(X = k) is the probability of getting exactly k successes
(n C k) is the number of ways to choose k successes from n trials (also known as the binomial coefficient)
p is the probability of success on a single trial
n is the number of trials
In this case, the rifleman averages 8 hits out of 10 shots, so the probability of hitting the target on a single shot is p = 8/10 = 0.8.
We want to find the probability of hitting the target in 3 out of 4 shots, so k = 3 and n = 4.
Using the binomial probability formula, we can calculate the probability as follows:
P(X = 3) = (4 C 3) * 0.8^3 * (1 - 0.8)^(4 - 3)
P(X = 3) = (4 C 3) * 0.8^3 * 0.2^1
P(X = 3) = (4 / 3!) * 0.8^3 * 0.2^1
P(X = 3) = (4 * 3 * 2 / 3! ) * 0.8^3 * 0.2^1
P(X = 3) = 4 * 0.8^3 * 0.2^1
Simplifying the calculation:
P(X = 3) = 4 * 0.512 * 0.2
P(X = 3) = 0.4096
Therefore, the probability that the rifleman will hit the target in 3 out of 4 shots is 0.4096 (or 40.96%).