Final answer:
To find the probability that at least 5 out of 100 runners did not finish the Boston Marathon, given a 97.4% completion rate, we use the binomial distribution. The correct probability after calculating the probabilities for 0 to 4 failures is 0.226, implying that the chance is about 22.6% for at least 5 runners not to finish the marathon.
Step-by-step explanation:
The question asks to find the probability that at least 5 out of 100 randomly chosen runners did not finish the marathon, given that 97.4% of runners complete the race. To solve this, note that the complement is the probability that fewer than 5 runners (0, 1, 2, 3, or 4 runners) did not finish the marathon. We will use the binomial probability formula:
P(X ≥ 5) = 1 - P(X < 5)
The binomial formula is:
P(X = k) = C(n, k) × p^k × (1-p)^(n-k)
where C(n, k) is the number of combinations of n items taken k at a time, p is the probability of success (in this case, the probability of not finishing), and k is the number of successes (runners not finishing).
Here, we compute the probabilities for k = 0, 1, 2, 3, and 4 and sum them up, subtracting the sum from 1 to get the probability of at least 5 failures:
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Each term is calculated using the binomial probability formula with p = 0.026 (since 2.6% did not finish, which is 1 - 0.974).
After calculating each term and summing them, we find that the probability of at least 5 not finishing is 0.226 (Option C).