(a)

Scenario: Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 10 of which have electrical defects and 15 of which have mechanical defects.

Problem: How many ways are there to randomly select 7 of these keyboards for a thorough inspection (without regard to order)?

Process: To find the number of combinations of keyboards in a population of keyboards, we need to use the combination formula (or a binomial).

Answer: 480,700.

Summary

The binomial or combination formula is used to find the number of combinations of objects you can make from a total population of objects.

  • - The size of the total population.
  • - The number of objects from the total population.

(b)

Problem: In how many ways can a sample of 7 keyboards be selected so that exactly two have an electrical defect?

Process: We need to find the number of combinations of 2 keyboards from our population of 10 keyboards with electrical defects and multiply that with the number of combinations of 5 keyboards from our population of 15 keyboards with mechanical defects.

Answer: 135,135.

(c)

Problem: If a sample of 7 keyboards is randomly selected, what is the probability that at least 6 of these will have a mechanical defect?

Process: To find the probability of at least six keyboards from our sample having a mechanical defect if we're randomly selecting from our population of 25 failed keyboards, 15 of which have a mechanical defect, we can use the hypergeometric probability formula.

Answer: 0.1175.

Summary

The hypergeometric probability formula is used to find the probability that a certain number of objects in a sample of size has a certain characteristic which is present in a population of objects out of a total population of objects.

  • - The size of the total population.
  • - The number of objects with a certain characteristic in the total population.
  • - The size of the sample being taken of the total population.
  • - The number of objects with the same characteristic in the sample.