factory received a shipment of 26 hammers, and the vendor who sold the items knows there are 5 hammers in the shipment that are defective. Before the receiving foreman accepts the delivery, he samples the shipment, and if too many of the hammers in the sample are defective, he will refuse the shipment.
For each of the following, give your responses as reduced fractions.
If a sample of 5 hammers is selected, find the probability that all in the sample are defective.
If a sample of 5 hammers is selected, find the probability that none in the sample are defective.
To find the probability that all hammers in the sample are defective, we can use the combinations formula.
The combinations formula is: C(n, k) = n! / (k!(n-k)!)
where n is the total number of items and k is the number of items to choose from the total.
1. The probability that all hammers in the sample are defective:
To find this probability, we will have to choose 5 defective hammers from 5 defective hammers out of a total of 26 hammers.
The combinations of choosing 5 defective hammers from 5 defective hammers is: C(5, 5) = 1
The combinations of choosing 5 hammers from a total of 26 hammers is: C(26, 5) = 65,780
The probability that all hammers in the sample are defective is thus: 1 / 65,780 = 1/65780
2. The probability that none of the hammers in the sample are defective:
In this case, we have to choose 5 non-defective hammers from 21 non-defective hammers out of a total of 26 hammers.
The combinations of choosing 5 non-defective hammers from 21 non-defective hammers is: C(21, 5) = 20,349
The probability that none of the hammers in the sample are defective is thus: 20,349 / 65,780 = 20349/65780