A local manufacturing company has a defect rate of 6% for its facility.
What is the probability that when five items are selected from the production line, AT LEAST ONE of them is defective?
To find the probability that at least one of the five selected items is defective, we can find the probability that none of the five items are defective and subtract it from 1.
The probability of an item being defective is 6% or 0.06.
The probability of an item not being defective is 1 - 0.06 = 0.94.
The probability that none of the five items are defective is calculated as follows:
P(no defective items) = P(item 1 not defective) * P(item 2 not defective) * P(item 3 not defective) * P(item 4 not defective) * P(item 5 not defective)
P(no defective items) = 0.94 * 0.94 * 0.94 * 0.94 * 0.94
P(no defective items) ≈ 0.8305
Therefore, the probability that at least one of the five selected items is defective is:
P(at least one defective item) = 1 - P(no defective items)
P(at least one defective item) ≈ 1 - 0.8305 ≈ 0.1695
The probability that at least one of the five selected items is defective is approximately 0.1695, or 16.95%.
To calculate the probability of at least one defective item out of five, we can use the complement rule.
The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.
In this case, we want to find the probability of at least one defective item, which is the complement of the probability that none of the items are defective.
First, let's find the probability that none of the items are defective. The probability of choosing a non-defective item from the production line is 1 - 0.06 = 0.94.
To find the probability that all five items selected are non-defective, we multiply this probability by itself five times:
P(non-defective) = 0.94 * 0.94 * 0.94 * 0.94 * 0.94 = 0.7350917376
Now, to find the probability of at least one defective item (the complement), we subtract the probability of none of the items being defective from 1:
P(at least one defective) = 1 - 0.7350917376 = 0.2649082624
Therefore, the probability that when five items are selected from the production line, at least one of them is defective is approximately 0.2649, or 26.49%.
To find the probability that at least one of the five selected items is defective, we need to calculate the complement of the event that none of the items is defective.
The probability of selecting a defective item is 6% or 0.06. So, the probability of not selecting a defective item is 1 - 0.06 = 0.94.
Since the selections are independent (the outcome of one does not affect the outcome of another), we can use the probability of not selecting a defective item to calculate the probability of none of the items being defective.
The probability of none of the five items being defective is found by multiplying the probability of not selecting a defective item by itself five times, since each selection is independent.
P(none defective) = 0.94 * 0.94 * 0.94 * 0.94 * 0.94 = 0.8305 (rounded to four decimal places)
Now, to find the probability of at least one defective item, we subtract this probability from 1.
P(at least one defective) = 1 - P(none defective) = 1 - 0.8305 = 0.1695 (rounded to four decimal places)
Therefore, the probability of selecting at least one defective item when five items are selected is approximately 0.1695 or 16.95%.