Predicting coincidence Your physical chemistry class has 19

Predicting coincidence. Your physical chemistry class has 19 students. What is the probability that at least two classmates have the same birthday?

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Solution

Your physical chemistry class has 19 students. What is the probability that at least two classmates have the same birthday?

The probability of at least one match includes 1 match, 2 matches, 3 matches, etc. Evaluating each probability would take a lot of work. To make it simpler, we can find the probability of at least one match by finding the complement probability of no matches.

Note:    P(no matches) + P(at least one match) = 1

P(at least one match) = 1 – P(no matches)

Suppose the class only has 2 students. The first student’s birthday could be any of the 365 days in a year. So the chance that the second student has a different birthday is 364/365. The probability that the two students have the same birthday is,

1-364/365 =1/365

Suppose the class has 3 students. The probability that all three have different birthdays is

P(no matches) = P(students 1, 2, and 3 have different birthdays)

= P(students 1 and 2 are different) x P(student 3 is different|students 1 and 2 are different)

The probability that student 3 has a different birthday from students 1 and 2 is 363/365 since there are 363 days left that are different from student 1 and 2.

=(364/365)*(363/365)

We can see that the method follows when we add more students to our classroom. Now consider 19 students in your class. Following the same method, by the time you arrive at the 19^th student, to have a different birthday he/she will have 18 less days to choose from out of 365. So the probability of the 19^th student having a different birthday from students 1 through 18 is 347/365.

P(no matches) = P(students 1 and 2 and 3 and 4… and 19 have different birthdays)

=(364/365)*(363/365)….(347/365)

The product of these probabilities equals 0.6513

P(at least one match) = 1 – P(no matches)

P(at least one match) = 1 – 0.6513

P(at least one match) = 0.3487