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 can use the concept of probability and the complement rule. Let's break it down step by step:
1. Define the event: Let's define the event A as "emission cloud movement in the critical direction" for a given launch.
2. Identify the probability of event A: We are given that the probability of emission cloud movement in the critical direction for any given launch is 5% or 0.05 (assuming this percentage is consistent).
3. Use the complement rule: The complement of event A is "no emission cloud movement in the critical direction." So, the probability of no movement in the critical direction is 1 - 0.05 = 0.95.
4. Calculate the probability of no movement in the critical direction for five launches in a row: Since the probability for a particular launch does not depend on other launches, we can assume independence. Therefore, the probability of no movement in the critical direction for all five launches is simply (0.95)^5.
5. Calculate the probability of at least one movement in the critical direction for five launches: The complement of "no movement in the critical direction for five launches" is "at least one movement in the critical direction for five launches." Using the complement rule again, this probability is 1 - (0.95)^5.
Hence, the answer is 1 - (0.95)^5 = 0.2262 or approximately 22.62%.