Computer processors are shipped in lots of 70 from a factory. Before being shipped, 21 are randomly tested from each lot. If any of these 21 fail, the entire lot is not shipped. What is the probability that a lot containing exactly 4 bad processors gets shipped?
To find the probability that a lot containing exactly 4 bad processors gets shipped, we need to determine the number of possible lots with 4 bad processors and then calculate the probability of one of those lots being shipped.
First, let's consider the number of ways to choose 4 bad processors from a lot of 70. This can be calculated using the combination formula, denoted as C(n, r), where n is the total number of items and r is the number of items to be selected. In this case, we have C(70, 4) ways to choose 4 bad processors from a lot of 70.
Next, we need to calculate the probability that all 21 tested processors from a lot are good. To do this, we first need to calculate the number of ways to choose 21 good processors from the remaining 66 processors (after removing the 4 bad processors). This can be calculated as C(66, 21).
The probability that all 21 tested processors from a lot are good is then given by the ratio of the number of favorable outcomes (in this case, choosing 21 good processors) to the total number of possible outcomes (choosing any 21 processors from the lot): C(66, 21) / C(70, 21).
Finally, the probability that a lot containing exactly 4 bad processors gets shipped is equal to 1 minus the probability that all 21 tested processors from the lot are good, since a lot with any bad processors will not be shipped. Therefore, the probability is 1 - (C(66, 21) / C(70, 21)).