Find the probability of at least one birthday match among a group of 48 people.
The probability is
1 - Probability of no matches
The probability of no matches is the ratio of the number of ways you can choose the birthdays such that each person has a different birthday (let's call this N) divided by the number of ways to distribute birthdays without restriction (let's call this M).
CLearly M = 365^48
and
N = 365!/(365 - 48)!
To see this, consider that the first person can have 365 birhtdays, the second can have 365-1 as his/her birthday bust be different from the first, The third can have 365 - 2 birthdays, etc. etc.
Then, to evaluate the numbers you have to be careful if your calculator does not display numbers larger than 10^100. You can then take logarithms and use Strirling's formula to evaluate the factorials, or simply sum the 48 terms in the expression for Log(N).
Yo should fiund that 1 - N/M = 0.9606
To find the probability of at least one birthday match among a group of 48 people, we can use the concept of complement probability.
Let's calculate the probability of no birthday matches among the 48 people first.
The first person can have any birthday without a match, so the probability is 365/365.
When the second person enters, there is a 364/365 probability that their birthday will not match the first person.
For the third person, there is a 363/365 probability that their birthday will not match either of the first two people.
Following this pattern, for the 48th person, there will be a 318/365 probability that their birthday will not match any of the previous 47 people.
To calculate the probability of no birthday matches, we multiply these probabilities together:
P(no matches) = (365/365) * (364/365) * (363/365) * ... * (318/365)
Now, to find the probability of at least one birthday match, we can subtract the probability of no matches from 1:
P(at least one match) = 1 - P(no matches)
P(at least one match) = 1 - [(365/365) * (364/365) * (363/365) * ... * (318/365)]
To calculate this probability, you can use a calculator or a programming language. The result will be the probability of at least one birthday match among a group of 48 people.
To find the probability of at least one birthday match among a group of 48 people, we can use the concept of the "Birthday Problem" or the "Birthday Paradox."
The probability can be calculated by considering the complement of the event (i.e., the probability of no birthday matches) and subtracting it from 1.
Let's break down the problem into steps:
Step 1: Determine the total number of possible combinations of birthdays for the group of 48 people.
Since there are 365 possible birthdays, each person can have any one of these birthdays. Therefore, the total number of combinations is given by:
365^48
Step 2: Calculate the number of combinations where there are no birthday matches.
For the first person, there are 365 possible birthdays to choose from. For the second person, there are 364 choices remaining (as one birthday is already taken by the first person). Similarly, the third person has 363 choices, the fourth person has 362 choices, and so on. Therefore, the total number of combinations with no birthday matches is:
365 * 364 * 363 * ... * (365 - 48 + 1)
Step 3: Calculate the probability of no birthday matches.
The probability of no birthday matches is given by dividing the number of combinations with no birthday matches (from Step 2) by the total number of possible combinations (from Step 1).
Probability of no birthday matches = [365 * 364 * 363 * ... * (365 - 48 + 1)] / 365^48
Step 4: Calculate the probability of at least one birthday match.
The probability of at least one birthday match is equal to 1 minus the probability of no birthday matches.
Probability of at least one birthday match = 1 - [365 * 364 * 363 * ... * (365 - 48 + 1)] / 365^48
Calculating this expression will give you the probability of at least one birthday match among a group of 48 people.