Chance of 23 people have atleast one shared birthday

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In summary, the conversation discusses the odds of at least two people having the same birthday when there are 23 people in a group. The calculation for this is 1 - (365P23 / 365^23), but the person speaking is unsure if their calculation is correct. The correct calculation may be 1 - (364! / (365^22 * 342!)), which suggests they may have used 364P22 instead of 365P23.
  • #1
Addez123
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TL;DR Summary
What's the odds that, when having 23 people gatherd, that atleast 2 has same birthdate?
The odds should be 1 - The odds of nobody having the same birthday
The odds of nobody having same birthday should be:

$$365/365 * 364/365 * 363/365 ... (365-22)/365 = 365!/(365-23)! = 365P23$$
However, $$365P23 = 4.22 * 10^58$$ so I'm obviously doing something wrong here but I can't see what.
 
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  • #2
Addez123 said:
Summary:: What's the odds that, when having 23 people gatherd, that atleast 2 has same birthdate?

The odds should be 1 - The odds of nobody having the same birthday
The odds of nobody having same birthday should be:

$$365/365 * 364/365 * 363/365 ... (365-22)/365 = 365!/(365-23)! = 365P23$$
However, $$365P23 = 4.22 * 10^58$$ so I'm obviously doing something wrong here but I can't see what.
Not multiplying correctly?
 
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  • #3
PeroK said:
Not multiplying correctly?
What part is incorrect?
365P23 = 42,200,819,302,092,400,000,000,000,000,000,000,000,000,000,000,000,000,000,000
I double checked.
 
  • #4
Addez123 said:
What part is incorrect?
365P23 = 42,200,819,302,092,400,000,000,000,000,000,000,000,000,000,000,000,000,000,000
I double checked.
All the numbers you are multiplying together are less than 1.
 
  • #5
You left out a factor ##\frac{1}{365^{23}}## in your first line summary.
 
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  • #6
Oh I see.
The $$\frac 1 {365^{23}}$$
is the total amount of possible birthdays for all people.

I encountered another issue tho. In my textbook, the answer is
$$1-364!/(365^{22} \cdot 342!) \approx .507$$
It seems here they've calculated 364P22 instead of 365P23.
I don't see why tho?
 
  • #7
Addez123 said:
Oh I see.
The $$\frac 1 {365^{23}}$$
is the total amount of possible birthdays for all people.

I encountered another issue tho. In my textbook, the answer is
$$1-364!/(365^{22} \cdot 342!) \approx .507$$
It seems here they've calculated 364P22 instead of 365P23.
I don't see why tho?
I suspect that 364 and 22 are the numbers for other people (22) not having the same birthday (364 not).
 

1. What is the probability of 23 people having at least one shared birthday?

The probability of 23 people having at least one shared birthday is approximately 50.7%. This is calculated using the formula 1 - (365!/365^23) which takes into account all the possible combinations of birthdays and subtracts it from 1 to get the probability of at least one shared birthday.

2. How does the probability change as the number of people increases?

The probability of at least one shared birthday increases as the number of people increases. For example, with 30 people, the probability increases to 70.6% and with 40 people, the probability increases to 89.1%. This is due to the fact that as the number of people increases, the number of possible combinations of birthdays also increases, making it more likely for at least one shared birthday to occur.

3. Is this phenomenon known as the "Birthday Paradox"?

Yes, this phenomenon is commonly known as the "Birthday Paradox" or "Birthday Problem". It is called a paradox because the probability of at least one shared birthday is higher than what most people would intuitively expect. It is not a true paradox, but rather a counterintuitive result based on the large number of possible combinations.

4. How does leap year affect the probability of shared birthdays?

Leap year does not significantly affect the probability of shared birthdays in a group of 23 people. This is because the extra day in a leap year only adds one more possible birthday combination, which does not significantly change the overall probability. However, in larger groups, the probability may be slightly higher due to the additional possible combinations.

5. Can this probability be applied to other scenarios besides birthdays?

Yes, this probability can be applied to any scenario where there is a limited number of options and a group of people. For example, it can be used to calculate the probability of two people sharing the same birth month, or the probability of two people having the same favorite color in a group of 23 people. However, it may not be applicable in scenarios where there is a larger number of options, such as a group of people sharing the same favorite number out of 100 possible numbers.

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