Birthday Problem

Suppose that you are in a group of 40 people. Would you bet for or against at least two of the people out of 40 having the same birthday?

Probability of at least two out of 40 having the same birthday can be estimated through a simulation.

Note that P(at least two have the same birthday)=1-P(none of them has the same birthday).

Follow the following steps to carry out the simulation to estimate the proabbaility that none of them has the same birthday:

• Step 1: Click on the "Generate" button on the Statlets window at the end of this page.
• Step 2: Click on the "Random numbers" button.
• Step 3: Click on the "Discrete Uniform" button & select the sample size. Discrete uniform random variable takes k different values with equal probability.Also type 40 in the "Sample size" box.
• Step 4: Click on the "OK" button
• Step 5: Type 1 and 365 in the "Lower limit" and "Upper limit" boxes, respectively. Here we are assuming that there are 365 days in a year and that all birthdays are equally likely. Click "OK". You will see the 40 simulated birthdays. To see the matches easily carry out the last two steps
• Step 6: Select the "Histogram"
• Step 7: Type 365 for "Number of classes" 1 for "Lower limit" 365 for "Upper limit". Click "OK". By using the histogram you can see how many matches you ended up with for each birthday.
• Repeat the Steps 1 through 7 ten times.
• Find your estimate by using (number of cases out of 10 that you ended up with at least one match)/10.

You can view these steps at the below animation.

The following table provides the theoretical probabilities of no two students having the same birthday when there are 1-40 people in the group. These probabilities are calculated by using the counting rules. The theoretical probability for at least two having the same birthday in a group of 30 students is 1-0.29368=0.70632.