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.
To answer these questions, we need to use the concept of probability.
1. A given launch will not result in cloud movement in the critical direction.
To calculate the probability of a launch not resulting in cloud movement in the critical direction, we can use the complement rule. The complement rule states that the probability of an event occurring is 1 minus the probability of that event not occurring.
Given that the weather conditions cause emission cloud movement in the critical direction only 5% of the time, the probability of a launch not resulting in cloud movement in the critical direction is 1 - 0.05 = 0.95 or 95%.
2. Any 5 launches will result in at least one cloud movement in the critical direction.
To find the probability that any 5 launches will result in at least one cloud movement in the critical direction, we can use the complementary probability again. We want to find the probability that none of the 5 launches result in cloud movement in the critical direction and then subtract that from 1 to get the probability of at least one cloud movement.
Since each launch is independent and has a 5% chance of cloud movement, the probability that none of the 5 launches have cloud movement in the critical direction is (0.95)^5 = 0.7738 or approximately 77.38%.
Therefore, the probability of at least one cloud movement in the critical direction in 5 launches is 1 - 0.7738 = 0.2262 or approximately 22.62%.