Final answer:
The probability that the archer gets exactly 4 bull's-eyes out of 9 shots, with each shot having a 46% chance of hitting the bull's-eye, is approximately 0.286.
Step-by-step explanation:
To find the probability that an archer gets exactly 4 bull's-eyes out of 9 shots when she hits the bull's-eye 46% of the time, we use the binomial probability formula:
P(X = k) = C(n, k) * (p)^k * (1-p)^(n-k)
- n = total number of shots (9)
- k = number of bull's-eyes we are looking for (4)
- p = probability of hitting the bull's-eye on a single shot (0.46)
First, we calculate C(n, k), which is the combination of n items taken k at a time:
C(9, 4) = 9! / (4! (9-4)!) = 126
Next, we calculate p^k and (1-p)^(n-k):
(0.46)^4 = 0.04488
(1-0.46)^(9-4) = (0.54)^5 = 0.04557
Now, we multiply them together to get the final probability:
P(X = 4) = 126 * 0.04488 * 0.04557 ≈ 0.286
The probability that she gets exactly 4 bull's-eyes out of 9 shots is therefore approximately 0.286.