Question

Generating the Birthday Probabilities Example 5 of this section concerns the probability that, in a group of \(n\) people, at least two people will share a common birthday. You can generate these probabilities on your calculator for values of \(n\) from 1 to \(365 .\) Step 1: Set the values of \(N\) and \(P\) to zero: Step \(2 :\) Type in this single, multi-step command: Now each time you press the ENTER key, the command will print a new value of \(N(\) the number of people in the room) alongside \(P\) (the probability that at least two of them share a common birthday): If you have some experience with probability, try to answer the following questions without looking at the table: (a) If there are three people in the room, what is the probability that they all have different birthdays? (Assume that there are 365 possible birthdays, all of them equally likely.) (b) If there are three people in the room, what is the probability that at least two of them share a common birthday? (c) Explain how you can use the answer in part (b) to find the probability of a shared birthday when there are four people in the room. (This is how the calculator statement in Step 2 generates the probabilities.) (d) Is it reasonable to assume that all calendar dates are equally likely birthdays? Explain your answer.

Short answer

a) The probability that all three have different birthdays: approximately 0.9918. b) The probability that at least 2 out of 3 share a birthday: approximately 0.0082. c) The probability for \(n\) people sharing a birthday is computed similarly, subtracting the product of nominal fractions from 1. d) The assumption of all calendar dates being equally likely for birthdays is an oversimplified model, and may deviate from reality where multiple other factors come into play.

Step-by-step solution

Calculate the probability that three people all have unique birthdays

Firstly, it's necessary to calculate the probability that the three people all have different birthdays. This can be evaluated by calculating the number of possible ways the three birthdays could be distinct and dividing it by the total number of possibilities. This is given by: \((365/365) * (364/365) * (363/365)\). Compute this expression.

Calculate the probability that at least two of them share a common birthday

The probability that at least two out of the three people share a common birthday is found by subtracting the probability of them all having unique birthdays (computed in step 1) from 1. This is computed as: \(1-[(365/365) * (364/365) * (363/365)]\).

Explain how to use part (b) to find the probability of a shared birthday when there are four people

You can generalize from part (b) to compute the probability for a group of four people. The probability that they all have unique birthdays is \((365/365) * (364/365) * (363/365) * (362/365)\), thus the probability at least two share a birthday is \(1 - (365/365) * (364/365) * (363/365) * (362/365)\). More generally, you can compute this for any number \(n\) of people by repeating this process, reducing the numerator of each new fraction by 1.

Discuss assumptions

It's important to acknowledge that reality might be a bit more complex than the mathematical model. All dates being equally likely for birthdays could be affected by several factors - seasons, holidays, hospital scheduling, amongst others - and this may cause some deviations from the theoretical model. In practice, this assumption is a simplification that makes the problem tractable.