Main Concept

Probabilities can sometimes be difficult to think about. For example, when asked the following question:

Given a room filled with N people, what is the probability that any pair of people in the room share a birthday?

Many people might try to simplify the problem, and ask themselves the easier question,

If I am in a room filled with N people, what is the probability that I share a birthday with someone else?

The answer to the second question is simply (N-1)/365 (disregarding February 29th and assuming a uniform distribution of birthdays). However, the answer to the first question is much different! For example, if there are 23 people in the room, the answer to the second question is only 6%, but (as we will see below) the answer to the first is roughly 50%. This is called the birthday paradox.

This demonstration allows you to investigate the birthday problem. You can enter birthdays one at a time by either making them up yourself or by letting Maple choose a random one. You can also let Maple fill the remaining entries in the table with random birthdays.

When the table is filled, Maple will check whether or not the table has any matches. If at least one match is found, it is indicated at the bottom of the table. The total number of tables with matches is shown in the bottom display area, as well as the updated percentage. The updated percentage can be compared to the theoretical probability each time the table is filled.

Explanation of the Probability

Since there are 365 days in the year, the probability of any two people not having the same birthday is $\frac{364}{365}$, the probability of a third person having a different birthday than the other two people is $\frac{363}{365}$, and so on. Thus, in a room containing 23 people, the probability of no two people sharing the same birthday will be:       $\frac{364}{365}·\frac{363}{365}·..\.·\frac{365-22}{365} \= \frac{364·363·..\.·\left(365-22\right)}{{365}^{22}}$                                                                            $\=\frac{365·364·363·..\.·343}{{365}^{23}}\=\frac{\frac{365\!}{342\!}}{{365}^{23}}\approx 0.5$ or $50$%.     If this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is simply $1-0.5\=0.5$ or 50%.

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MathApps/ProbabilityAndStatistics