Investigation 4: Johnny Carson’s Birthday (due Thursday, September 30)

You may work with one other person on this assignment, handing in one report with both names. Word-processed reports are preferred to hand-written ones.

Legend has it that a guest on Johnny Carson’s Tonight show, aware of the “birthday problem,” told Johnny that there was almost a certainty of shared birthdays in his audience of over 100 people. Johnny allegedly asked the audience, “Was anyone else born on my birthday- October 23?” and found that nobody was.

a) Explain how Johnny’s question differs from the one that we analyzed in class.

b) Determine expression for the probability that at least one person in a group of n shares Johnny’s birthday. [Hint: Use the same simplifying assumptions as before, and again use the complement rule.]

c) Evaluate this probability for the following values of n: 5, 10, 20, 30, 50, 75, 100. [Hint: Feel free to use Excel. You may need to simplify the expression before entering it into Excel.]

d) Produce a graph of this probability as a function of n. Comment on the function’s behavior.

e) Describe how this function compares to the one involving the probability of matching any birthday.

f) Identify the smallest value of n for which the probability of matching Johnny’s birthday exceeds .5. What is this probability?

g) Repeat f) for the probability to exceed .9.

h) Is there a value of n for which the probability of at least one person matching Johnny’s birthday equals one (exactly)? Explain.