Suppose a planet has 13 months in a year What is the probabi

Suppose a planet has 13 months in a year. What is the probability that in a group of 5 people on this planet, at least two of them will have a birthday in the same month? What is the probability that in a group of 8 people on this planet, at least two of them will have a birthday in the same month? What is the probability that in a group of 13 people on this planet, at least two of them will have a birthday in the same month? What is the probability that in a group of 15 people on this planet, at least two of them will have a birthday in the same month?

Solution

I will explain the first question.

Well, the first person has 4 comparisons to make, but the second person was already compared to the first person, so there are only 3 comparisons to make. The third person then has 2 comparisons, the fourth person has 1. If you add up all possible comparisons (4 + 3 + 2 + 1) the sum is 10 comparisons, or combinations.

We\'ll suppose that the year has 30*13=390 days

The probability that a person does not have the same birthday as another person is 12 divided by 13 because there are 12 monts that are not a person\'s birthday. This means that any two people have a 12/13, or 92.3077 percent, chance of not matching birthdays.

If you multiply calculate (12/13)¹⁰, you\'ll find there\'s a 44.91 percent chance that all 10 comparisons contain no matches.

Consequently, the odds that there is a birthday match in those 10 comparisons is 1 – 0.4491 percent = 0.5509=55.09 percent

If we do the same with 13 people, we have:

(12+11+...+1)=78

The probability that at least two of them will have a birthday in same month is: 1-(12/13)²⁸=0.9981=99.81 percent

For 15 person the probability that at least two of them will have a birthday in same month is 100%

If we do the same with 8 people, we have:

(7+6+5+...+1)=28

The probability that at least two of them will have a birthday in same month is: 1-(12/13)²⁸=0.8937=89.37 percent