There were 42 different presidents of the United States from 1789 through 2000. What is the probability that at least two of them had the same birthday?
To find the probability that at least two presidents had the same birthday out of the 42 different presidents, we can use the concept of complementary probability.
Step 1: Define the event of interest.
Let A be the event that at least two presidents had the same birthday.
Step 2: Identify the total number of possible outcomes.
Since there are 365 days in a year (ignoring leap years), the total number of possible birthdays for the presidents is 365.
Step 3: Calculate the number of favorable outcomes.
The number of favorable outcomes is the number of ways we can have at least two presidents with the same birthday.
To calculate this, we will use the complement of event A, denoted as A'.
Step 4: Calculate the complement of event A.
To find A', we need to calculate the probability that all the presidents have unique birthdays.
For the first president, there are 365 possible birthdays.
For the second president, there are 364 possible birthdays (since one day is already taken by the first president).
For the third president, there are 363 possible birthdays (since two days are already taken by the previous presidents).
Continuing this pattern, the number of possible birthdays for the 42nd president would be 324.
So, the number of favorable outcomes for event A' is 365 * 364 * 363 * ... * 324.
Step 5: Calculate the probability of event A.
Now we can calculate the probability of event A by taking the complement of event A', that is:
P(A) = 1 - P(A')
Step 6: Calculate the probability.
To find the probability, we need to divide the number of favorable outcomes by the total number of possible outcomes.
P(A) = 1 - (365 * 364 * 363 * ... * 324) / 365^42
Calculating this expression will give you the probability that at least two presidents had the same birthday out of the 42 different presidents.
To find the probability that at least two of the 42 different US presidents had the same birthday, we can use a technique called the "Birthday Problem" or "Birthday Paradox".
The Birthday Problem involves determining the probability that in a group of n people, there exists at least one pair of individuals with the same birthday. In this case, we are considering 42 individuals.
To solve this problem, we can calculate the probability of no two presidents having the same birthday and then subtract it from 1 to get the probability of at least two presidents having the same birthday.
Here's how you can calculate it step by step:
1. Calculate the probability of no two presidents sharing the same birthday:
- The first president can have any birthday (365/365 probability).
- The second president should have a different birthday than the first one (364/365 probability).
- The third president should have a different birthday than the first two (363/365 probability).
- Continuing this pattern, the nth president should have a different birthday than the previous n-1 presidents (365 - (n-1))/365 probability.
2. Multiply all the probabilities together to get the probability of no two presidents sharing the same birthday.
3. Subtract the probability of no two presidents sharing the same birthday from 1 to get the probability that at least two presidents have the same birthday.
It's important to note that this calculation assumes that each day of the year (365 days) is equally likely to be the birthday of a president, which is a simplification.
Let me calculate the exact probability for you...
about .5
https://en.wikipedia.org/wiki/Birthday_problem
1 - [(1/365)^41 (364*363*362....324)]
is the difference of Chrissy's height and
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