Given 12 people, what is the probability that at least two people would have the same birth-month? Given 13 people, what is the probability that at least two people would have the same birth-month?
To calculate the probability of at least two people having the same birth-month, we can use the principle of complementary probability.
First, let's calculate the probability that all 12 (or 13) people have different birth-months.
For the first person, there are 12 (or 13) possible birth-months they could have.
For the second person, there are 11 (or 12) remaining birth-months to choose from, as one birth-month has already been assigned to the first person.
Similarly, for each subsequent person, the number of remaining birth-months decreases by 1.
So, the probability that all 12 (or 13) people have different birth-months is:
(12/12) * (11/12) * (10/12) * ... * (1/12)
or
(13/13) * (12/13) * (11/13) * ... * (1/13)
To find the probability of at least two people having the same birth-month, we subtract this probability from 1, as the complementary event represents the case where at least two people have the same birth-month.
Therefore, the probability of at least two people having the same birth-month given 12 people is:
1 - (12/12) * (11/12) * (10/12) * ... * (1/12)
Similarly, the probability of at least two people having the same birth-month given 13 people is:
1 - (13/13) * (12/13) * (11/13) * ... * (1/13)
Calculations:
For 12 people:
P(at least two people have the same birth-month) = 1 - (12/12) * (11/12) * (10/12) * ... * (1/12)
= 1 - 0.3229
= 0.6771 (approximately)
For 13 people:
P(at least two people have the same birth-month) = 1 - (13/13) * (12/13) * (11/13) * ... * (1/13)
= 1 - 0.3710
= 0.6290 (approximately)
Therefore, the probability is approximately 0.6771 for 12 people and 0.6290 for 13 people.