Question

Five friends-Allison, Beth, Carol, Diane, and Evelyn- have identical calculators and are studying for a statistics exam. They set their calculators down in a pile before taking a study break and then pick them up in random order when they return from the break. What is the probability that at least one of the five gets her own calculator? [Hint: Let A be the event that Alice gets her own calculator, and define events B, C, D, and E analogously for the other four stu- dents.] How can the event (at least one gets her own calcu- lator} be expressed in terms of the five events A, B, C, D, and E? Now use a general law of probability. [Note: This is called the matching problem. Its solution is easily extended to individuals. Can you recognize the result when n is large (the approximation to the resulting series)?]​

Answer 1

The probability that at least one of the five friends gets her own calculator can be expressed in terms of the events A, B, C, D, and E as follows:

P(at least one gets her own calculator) = 1 - P(none of them get their own calculator)

To calculate P(none of them get their own calculator), we can use the principle of inclusion-exclusion:

P(none of them get their own calculator) = 5!/(5-5)!*(0!)- 4!/(5-4)!*(1!)+ 3!/(5-3)!*(2!)- 2!/(5-2)!*(3!)+ 1!/(5-1)!*(4!)
= 1 - 1 + 10/120 - 15/120 + 24/120
= 8/15

Therefore, the probability that at least one of the five friends gets her own calculator is:

P(at least one gets her own calculator) = 1 - P(none of them get their own calculator)
= 1 - 8/15
= 7/15