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 item out of five is defective, it is easier to compute the probability that all five items are non-defective and then subtract it from 1.
Assuming each item has an independent probability of being defective, let's say the probability of an item being defective is p and the probability of an item being non-defective is q (where q = 1 - p).
The probability of selecting a non-defective item is q, and since there are five items, the probability of selecting all five as non-defective is q * q * q * q * q = q^5.
Therefore, the probability of at least one item being defective is 1 - q^5.
Note: If the probability of an item being defective (p) is not given in the problem, you would need that information to calculate the exact probability.
To calculate the probability that at least one item is defective out of five selected, we can use the complement rule:
1. Find the probability that none of the items is defective.
2. Subtract this probability from 1 to get the probability that at least one item is defective.
Let's break down the steps:
Step 1: Find the probability that none of the items is defective.
Assuming the probability of an item being defective is p, the probability of an item not being defective (q) is 1 - p.
The probability of selecting a non-defective item on each pick is q. Since we are selecting five items without replacement (meaning that once an item is selected, it is not put back into the line), the probabilities multiply together.
So, the probability of selecting five non-defective items in a row is q * q * q * q * q, or q^5.
Step 2: Subtract this probability from 1 to get the probability that at least one item is defective.
The probability of at least one defective item is then 1 - q^5.
Therefore, the probability that when five items are selected from the production line, at least one of them is defective is 1 - q^5.
Note: To calculate the exact probability, the value of p (probability of an item being defective) would be needed.
To find the probability that at least one item is defective when five items are selected, we can calculate the complementary probability, which is the probability that no items are defective, and subtract it from 1.
To calculate the probability that no item is defective in a single selection, we divide the number of non-defective items by the total number of items. Let's assume there are 100 items in total, with 90 non-defective items and 10 defective items.
The probability of selecting a non-defective item in a single selection is 90/100 = 0.9. Therefore, the probability of not selecting a defective item in a single selection is also 0.9.
Since we are selecting five items, we can calculate the probability of not selecting a defective item in any of the five selections by multiplying the probabilities together:
(0.9)^5 ≈ 0.59049
This is the probability that none of the selected items are defective. To find the probability that at least one item is defective, subtract this value from 1:
1 - 0.59049 ≈ 0.40951
Therefore, the probability that at least one item is defective when five items are selected is approximately 0.40951, or 40.951%.