One problem encountered by developers of the space shuttle program is air pollution in the area surronding the launch site. Acertain direction from the launch site is considered critical in terms of hydrogen chloride pollution from the exhaust cloud. It has been determined that weather conditions would cause emission cloud movement in the critical direction only 5% of the time. Assume that probabilities for a particular launch in no way depend on the probabilities for other launches. A given launch will not result in cloud movement in the critical direction. Any 5 launches will result in at least one cloud movement in the critical direction.
To solve this problem, we need to determine the probability of having at least one cloud movement in the critical direction out of five launches.
Let's start by finding the probability of not having any cloud movement in the critical direction for a single launch. We know that the probability of cloud movement in the critical direction is 5%, which implies that the probability of no cloud movement is 1 - 0.05 = 0.95.
Next, we can calculate the probability of not having any cloud movement in the critical direction for all five launches. Since the launches are independent events, we multiply the probabilities of not having cloud movement for each launch:
P(No cloud movement for one launch) = 0.95
P(No cloud movement for five launches) = (0.95)^5 ≈ 0.7738
The probability we just calculated is the probability of no cloud movement in any of the five launches. To find the probability of having at least one cloud movement, we subtract this value from 1:
P(At least one cloud movement in five launches) = 1 - P(No cloud movement for five launches)
P(At least one cloud movement in five launches) = 1 - 0.7738
P(At least one cloud movement in five launches) ≈ 0.2262
Therefore, the probability of having at least one cloud movement in the critical direction out of five launches is approximately 0.2262 or 22.62%.