In group of three friends what is the probability that at least two have the same birthday?
The first friend will have the birthday on some given day, doesn't matter when.
The prob that the second friend will have a different birthday = 364/365
the prob that the third friend will have a birthday from the other two = 363/365
prob that all three have different birthdays
= 1(364/365)(363/365)
So prob that somebody has the same birthday
= 1 - the above
= appr .0082
To calculate the probability that at least two friends in a group of three have the same birthday, we can use the complementary probability approach.
Step 1: Calculate the probability that all three friends have different birthdays.
The first person can have any birthday of 365 days. The second person must have a different birthday, so there are 364 possible choices. Similarly, the third person must have a different birthday from the first two, so there are 363 possible choices. Therefore, the probability that all three friends have different birthdays is:
P(all different) = (365/365) * (364/365) * (363/365)
Step 2: Calculate the probability that at least two friends have the same birthday using the complementary probability.
The complementary probability is the probability of the event not happening. Therefore, the probability that at least two friends have the same birthday is:
P(at least two same) = 1 - P(all different)
Let's calculate it:
P(at least two same) = 1 - [(365/365) * (364/365) * (363/365)]
P(at least two same) ≈ 0.0082
So, the probability that at least two friends in a group of three have the same birthday is approximately 0.0082, or 0.82%.
To determine the probability that at least two friends in a group of three have the same birthday, we can break it down into two scenarios:
Scenario 1: All three friends have the same birthday.
Scenario 2: Two friends have the same birthday, while the other friend has a different birthday.
Let's calculate the probability for each scenario:
Scenario 1:
The probability that all three friends have the same birthday is the same as the probability of any one person having a specific birthday. There are 365 possible birthdays, so the probability of all three friends having the same birthday is 1/365.
Scenario 2:
In this scenario, first, we need to choose the birthday that the two friends share. There are 365 options for this. Then, the remaining friend can have any of the other 364 possible birthdays. Therefore, the probability for this scenario is (365 * 364) / (365^2).
Now, we add the probabilities of both scenarios to get the overall probability:
Overall Probability = Probability of Scenario 1 + Probability of Scenario 2
Overall Probability = (1/365) + (365 * 364) / (365^2)
Simplifying this expression will give you the final answer.