### Probability Calculations

In this lab you will compute probabilities using Excel functions for combinations and permutations. Of course, the real problem is to set up the calculations.

The functions we use are:
=FACT(n) to compute n!
=PERMUT(n,k) to compute P(n,k), the number of ways to order k of n objects, and
=COMBIN(n,k) to compute C(n,k), the number of ways to select a k-object subset from n objects.

Problem 1. In this problem you will practice using the counting functions listed above and verify relationships between them.

A. Enter 6 in cells A3 and 2 in cell B3 to serve as values for n and k. In cell A4 enter the cell formula =PERMUT(A3,B3). In cell B4 enter the formula =FACT(A3)/FACT(A3-B3). Change the values in cells A3 and B3 to 10 and 4 respectively to see what happens. In a text box explain why the two values in cells A4 and B4 are always the same.

B. In cell A5 enter the cell formula =COMBIN(A3,B3) and in cell B5 enter the formula =FACT(A3)/(FACT(A3-B3)*FACT(B3)). Explain in the text box created in Part A why the two values in cells A5 and B5 are always the same.

Problem 2. In this problem we compute the probabilities of the different types of matches associated with gaming problems like the 1-key Bandit game.

A. For the 1-key Bandit problem we calculate the probabilities of three matches, two matches, and no matches (the only possibilities).
Sample Space size. Each of the three cells output in the game contain a fruit name randomly selected from among 6 names. Compute the size of the sample space and label it.
Three names match. If all three names agree there are exactly 6 possible results (one for each name from the pool). Compute the probability that all three names are the same.
Two names match. We count the number of ways that two of the three names are the same and the third is different by first selecting two of the three positions to contain the same names, then selecting one of the 6 names for these positions and one of the remaining 5 names for the final position. Compute this number; then the probability of a matching pair. Be sure to label the numbers.
No matching names. First count the number of ways to order 3 of the 6 names, then compute the probability. Be sure to label your values.

One way to check your calculations is to see if the sum of the three probabilities is 1.

B. Consider a modified version of the 1-key Bandit game which has 10 fruit names and each play randomly selects 4 names. Compute the probabilities for each of the possibilities in the following table.

 Outcomes Probabilities 4 names match 3 names match 2 matching pairs one matching pair No matches
Think carefully about how to compute the number of way a match can occur. Be sure to present your answers clearly.

Problem 3. A Math 105 class has 23 students. Compute the probability that at least two students in class have the same birthday. Assume 365 days in a year. In a text box explain the strategy used to compute the probability.