Chance of 23 people have atleast one shared birthday

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Discussion Overview

The discussion revolves around calculating the probability that at least two people in a group of 23 share the same birthday. Participants explore the mathematical formulation of the problem, including the correct application of permutations and the total number of possible birthdays.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes that the odds of at least two people sharing a birthday can be calculated as 1 minus the odds of nobody sharing a birthday, using the formula $$365/365 * 364/365 * 363/365 ... (365-22)/365$$.
  • Another participant questions the multiplication process, suggesting that there may be an error in the calculations leading to an extremely high value for $$365P23$$.
  • A participant asserts that the value of $$365P23$$ is indeed correct as calculated, providing a large numerical result.
  • Another participant points out that a crucial factor, $$\frac{1}{365^{23}}$$, was omitted in the initial calculations, which represents the total number of possible birthday combinations.
  • One participant expresses confusion regarding a discrepancy with their textbook, which presents a different formula involving $$364P22$$ and suggests that this may relate to the number of people not sharing a birthday.

Areas of Agreement / Disagreement

Participants do not reach consensus on the correct approach to calculating the probability, with multiple views on the application of permutations and the factors involved in the calculations. Discrepancies in textbook references further complicate the discussion.

Contextual Notes

Some participants highlight potential missing factors in their calculations and express uncertainty about the correct interpretation of the problem, particularly regarding the use of permutations and the total number of birthday combinations.

Addez123
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TL;DR
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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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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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.
 
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.
 
You left out a factor ##\frac{1}{365^{23}}## in your first line summary.
 
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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?
 
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).
 

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