How many friends must you have to guarantee that at least five of them will have birthdays in the same month?

p(at least 5 have the same month)

p(someone shows w/ at least someone else)

P(s)+ P(d) = 100 %
100%- P(d)
(12 *11*10*9*8)/12^5
P(d) =95040/248832 = .3819
100% -0.3819 = 0.6181

61.81%

To guarantee that at least five friends have birthdays in the same month, we need to determine the minimum number of friends required.

Since there are 12 months in a year, each person can have their birthday in any of these 12 months.

To find the minimum number of friends required, we must consider the worst-case scenario. The worst-case scenario is that each of the first four friends has a birthday in a different month. After adding these four friends, any additional friend will ensure that at least five friends have birthdays in the same month.

So, the minimum number of friends required is 4 + 1 = 5.

Therefore, you must have at least five friends to guarantee that at least five of them have birthdays in the same month.

To guarantee that at least five friends will have birthdays in the same month, we need to determine the minimum number of friends required to make this happen.

First, let's understand the worst-case scenario for this situation. That would be if each friend's birthday fell in a different month. In that case, we would need 12 friends to cover every month of the year.

However, we are looking for a situation where at least five friends have birthdays in the same month. To achieve this, we can calculate the complement of the probability of no five friends having the same birth month.

To calculate this, we need to consider the number of ways we can choose five friends out of the total number of friends with their birthdays distributed across the 12 months.

If we have only four friends, they can have their birthdays in any month, giving us 12^4 possible combinations. Similarly, for the fifth friend, we have 12 possible months to choose from.

Therefore, we would need the number of combinations for choosing five friends out of the total number of friends to be greater than or equal to the total number of combinations for the worst-case scenario (12^5).

Let's calculate this:
12^5 = 248,832

Next, we need to find the minimum number of friends required to exceed this number of combinations. We can try increasing the number of friends until we reach this point.

Let's calculate the combinations for different numbers of friends and find the point at which it becomes greater than or equal to 248,832:

For six friends: 12^6 = 2,985,984 (more than 248,832)
For seven friends: 12^7 = 35,831,808 (more than 248,832)
For eight friends: 12^8 = 429,981,696 (more than 248,832)

Therefore, we would need a minimum of eight friends to guarantee that at least five of them have birthdays in the same month.

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