Final answer:
The expected number of bullseyes is 12 ⋅ 0.86 = 10.32. The probability of various outcomes for an Olympic archer shooting 12 arrows with an 86% accuracy rate is calculated using methods such as the binomial probability formula. The expected number of bullseyes can be found by multiplying the number of attempts by the probability of hitting the bullseye on a single shot.
Step-by-step explanation:
In probability, the probability of an event happening is the number of favorable outcomes divided by the total number of possible outcomes. In this case, the archer hits the bullseye 86% of the time and each shot is independent, so the probability of hitting the bullseye on a single shot is 0.86. Let's calculate the probabilities:
A) To find the probability that she never misses, we take the probability of hitting the bullseye on all 12 shots, which is (0.86)^12 ≈ 0.1547, or approximately 15.47%.
B) The probability of getting exactly 11 bullseyes can be calculated using the binomial probability formula. Since the probabilities are the same for all the shots, the formula becomes: ⋅ (0.86)^11 ⋅ (1-0.86)^(12-11) * C(12, 11) ≈ 0.3655, or approximately 36.55%.
C) To find the probability of getting between 6 and 10 bullseyes, we need to add up the probabilities of getting 6, 7, 8, 9, or 10 bullseyes. This can also be calculated using the binomial probability formula for each individual case, and then summing them up. The probability is approximately 72.92%.
D) To find the probability of getting less than 8 bullseyes, we need to calculate the probabilities of getting 0, 1, 2, 3, 4, 5, 6, or 7 bullseyes, and then sum them up. The probability is approximately 54.29%.
E) To find the probability of getting at least 8 bullseyes, we can find the complement of getting less than 8 bullseyes, which is 1 - 0.5429 = 0.4571, or approximately 45.71%.
F) To find the probability of getting at most 9 bullseyes, we can calculate the probabilities of getting 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 bullseyes, and then sum them up. The probability is approximately 89.20%.
G) To find the probability of getting between 9 and 12 bullseyes, inclusive, we need to calculate the probabilities of getting 9, 10, 11, or 12 bullseyes, and then sum them up. The probability is approximately 18.19%.
The expected number of bullseyes is calculated by multiplying the number of attempts (12) by the probability of hitting the bullseye on a single shot (0.86). The expected number of bullseyes is 12 ⋅ 0.86 = 10.32.