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. First question- A given launch will not result in cloud movement in the critical direction. Second question Any 5 launches will result in at least one cloud movement in the critical direction.

Question

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. First question- A given launch will not result in cloud movement in the critical direction. Second question Any 5 launches will result in at least one cloud movement in the critical direction.

Answer 1

To answer these questions, we need to apply the concept of probability. Let's break them down step by step:

Question 1: A given launch will not result in cloud movement in the critical direction.

For this question, we are given that the weather conditions will cause emission cloud movement in the critical direction only 5% of the time. Therefore, the probability of the cloud not moving in the critical direction during a given launch is 1 - 0.05 = 0.95. This means that there is a 95% chance that the cloud will not move in the critical direction.

Question 2: Any 5 launches will result in at least one cloud movement in the critical direction.

To answer this question, we need to calculate the probability of having at least one cloud movement in the critical direction in a sequence of 5 launches.

The probability of cloud movement in the critical direction during a single launch is 0.05 (as given). Therefore, the probability of no cloud movement during a single launch is 1 - 0.05 = 0.95.

To find the probability of no cloud movement in any of the 5 launches, we need to multiply the probabilities together: 0.95 * 0.95 * 0.95 * 0.95 * 0.95 = 0.95^5.

However, we are interested in the probability of at least one cloud movement in the critical direction. So, subtracting the probability of no cloud movement from 1 will give us the desired result:

1 - 0.95^5 ≈ 0.2262

Therefore, there is approximately a 22.62% chance that any sequence of 5 launches will result in at least one cloud movement in the critical direction.