Chance of 23 people have atleast one shared birthday

[Addez123]

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.

 

[PeroK]

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?

 

[Addez123]

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.

 

[PeroK]

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.

 

[mathman]

You left out a factor $\frac{1}{365^{23}}$ in your first line summary.

 

[Addez123]

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?

 

[mathman]

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).