First of all, make a guess as to how likely this is. Pick a number between 0 and 1 to represent your guess as to the probability that a group of N people have a common birthday among two of them. Next, to simulate testing several groups of N people, each person is to randomly pick a day of the year 20 times. To do this, first randomly pick a month by rolling a dodecahedral die. Use the result to pick the month. Next, flip a coin 5 times. Label heads as 1 and tails as 0. You then get a 5 digit number from this; for instance, if your flips are 0, 1, 0 ,1, 1, write this as 01011. This number corresponds to a number between 0 and 32 if you view it as the base 2 representation of a number. Write down the number this sequence represents, and use that to be the day of the month. If you get an invalid date, do it again. You may speed up the selection of your dates by finding all of the months first, then finding all of the days of the month.
An alternative way to randomly select a day of the month is to use two dice. By rolling the dice, you will get two numbers a,b. You can then obtain a date from the number 6(a-1) + b. There are a few ways to get an invalid date; if this happens roll the dice again.
Once everybody has randomly selected 20 dates, we will determine how many times out of 20 that there is a common birthday in each set of dates.
Problem 1. Give an estimate of the probably that a group of N people have a common birthday based on the outcome of the class experiment.
Problem 2. Open the program Maple in the computer lab, and then open the file Birthdays.mws. Once there, click on Edit, then Execute, then Worksheet. Simulate what the class did by entering at a Maple prompt (>) the command bdays(N); where N is the number of people in the class. Make sure to end the command with a semicolon. Do this twenty times and note how many times there is a common birthday. Finally, enter prob(N); to see what is the theoretical probability that there is a common birthday among N people.