# How do I finish this math problem?

So, I managed to start the problem and got somewhere. I think what I've done so far is at least right.
The problem goes as following:
How big does a group have to be for the probability that at least 2 in the group share the same birthday surpasses 50 percent?
So to start, I assumed that there are n people...
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So, I managed to start the problem and got somewhere. I think what I've done so far is at least right.

The problem goes as following:

How big does a group have to be for the probability that at least 2 in the group share the same birthday surpasses 50 percent?

So to start, I assumed that there are n people in the group. And the chance of at least 2 sharing a birthday is 1 minus the chance that no one shares a birthday.

Calculating the probability that no one shares a birthday, the first person would be allowed to have 365 out of 365 possible birthdays, and the second person is allowed to have 365-1=364 birthdays out of 365 since 1 birthday is already taken. So the probability will be

(365/365) * (364/365) * (363/365) * .... * (365-n+1/365) for n people, or simplified: (365! / ((365^n)(365-n)!))

and the probability of there existing overlapping birthdays is 1 - (365! / ((365^n)(365-n)!)) and putting it greater that 0.5 gives us

1- (365! / ((365^n)(365-n)!)) > 0.5 or that (365! / ((365^n)(365-n)!)) < 0.5

But I am stuck here and have no idea whatsoever as how to solve for the inequality. Been trying for an hour to get anywhere. Does anyone have any ideas?

The answer to the problem is supposed to be 23, so the equation should give an answer that is close to 23.

The problem goes as following:

How big does a group have to be for the probability that at least 2 in the group share the same birthday surpasses 50 percent?

So to start, I assumed that there are n people in the group. And the chance of at least 2 sharing a birthday is 1 minus the chance that no one shares a birthday.

Calculating the probability that no one shares a birthday, the first person would be allowed to have 365 out of 365 possible birthdays, and the second person is allowed to have 365-1=364 birthdays out of 365 since 1 birthday is already taken. So the probability will be

(365/365) * (364/365) * (363/365) * .... * (365-n+1/365) for n people, or simplified: (365! / ((365^n)(365-n)!))

and the probability of there existing overlapping birthdays is 1 - (365! / ((365^n)(365-n)!)) and putting it greater that 0.5 gives us

1- (365! / ((365^n)(365-n)!)) > 0.5 or that (365! / ((365^n)(365-n)!)) < 0.5

But I am stuck here and have no idea whatsoever as how to solve for the inequality. Been trying for an hour to get anywhere. Does anyone have any ideas?

The answer to the problem is supposed to be 23, so the equation should give an answer that is close to 23.

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