What is the probability that this year's graduation will fall on the birthday of exactly one of the 336 Seniors? What is the probability that there is more than one such Senior?
Well, let me put on my graduation cap and crunch some numbers for you. Now, assuming each Senior has an equally likely chance of having their birthday on graduation day, we have 336 potential birthday buddies.
For the probability that exactly one Senior has their birthday on graduation day, we have to choose one out of the 336 Seniors (because if more than one Senior had their birthday on that day, it wouldn't fall under this category anymore). So the probability would be 1 out of 336, which simplifies to a whopping 0.00297619047.
Now, for the probability that there is more than one Senior celebrating their birthday on graduation day, things get a little more complicated. We have to consider all possible combinations of Seniors, two or more. This involves some heavy combinatorial math that would make even a clown's head spin.
So, instead of getting all math-y on you, let me just say that the probability of multiple Senior birthdays landing on graduation day would depend on the number of Seniors with birthdays on that specific day. But hey, don't worry too much about it! Just remember to bring enough birthday cake for all potential doppelgangers.
To find the probability that this year's graduation will fall on the birthday of exactly one of the 336 seniors, we need to calculate the probability that one senior has their birthday fall on the graduation day and all the other seniors do not.
Step 1: Calculate the probability that one specific senior has their birthday on the graduation day:
The probability of one specific senior having their birthday on a specific day is 1/365 (assuming the year has 365 days).
Step 2: Calculate the probability that the other 335 seniors do not have their birthday on the graduation day:
The probability that one specific senior does not have their birthday on the graduation day is 364/365.
Step 3: Calculate the probability that exactly one senior has their birthday on the graduation day:
To calculate this probability, we multiply the probability from Step 1 by the probability from Step 2, raised to the power of the number of other seniors, which is 335.
Probability = (1/365) * (364/365)^335
To find the probability that there is more than one senior with their birthday on the graduation day, we can subtract the probability we just calculated from the total probability. The total probability is the complement of the probability that none of the seniors have their birthday on the graduation day.
Step 4: Calculate the probability that none of the 336 seniors have their birthday on the graduation day:
The probability that one senior does not have their birthday on the graduation day is 364/365, so the probability that none of the 336 seniors have their birthday on the graduation day is (364/365)^336.
Step 5: Calculate the probability that more than one senior has their birthday on the graduation day:
Probability = 1 - (364/365)^336
Now you have the probability that this year's graduation will fall on the birthday of exactly one senior and the probability that more than one senior will have their birthday on the graduation day.
Note: These calculations assume that each senior's birthday is equally likely to fall on any given day of the year, and the probability of being born on any given day is the same for all seniors.
To calculate the probability, we need to know the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes is the number of days in a year, which is 365 (assuming it's not a leap year).
Now let's consider the favorable outcomes. To find the probability that this year's graduation will fall on the birthday of exactly one senior, we need to choose one senior out of 336 and then calculate the probability that their birthday matches the graduation day.
The probability that a specific senior's birthday matches the graduation day is 1/365 (since there is one specific day out of 365 that matches their birthday).
Since we need exactly one senior to have a matching birthday, we need to choose one senior out of the 336 available. This can be done in 336 different ways.
Therefore, the probability that this year's graduation will fall on the birthday of exactly one of the 336 seniors is (336/365) * (1/365) = 0.00079 (approximately).
To calculate the probability that there is more than one senior with a matching birthday, we can use the complement rule. The complement of "more than one senior" is "zero or one senior."
The probability that there is zero senior with a matching birthday is (364/365)^336 (since there are 364 days out of 365 that do not match their birthday).
The probability that there is exactly one senior with a matching birthday is 336 * (1/365) * (364/365)^335 (since we have to choose one senior out of 336 and the remaining 335 seniors should not have matching birthdays).
Finally, we can subtract these probabilities from 1 to get the probability of more than one senior having a matching birthday:
Probability of more than one senior = 1 - (364/365)^336 - 336 * (1/365) * (364/365)^335
This calculation will give you the probability of more than one senior having a birthday that matches the graduation day.