Question

_{Suppose n unrelated families (defined as the husband, the wife, and one child) are gathered together. What is the smallest n for which chances are > 50% that there will be two or more families completely matched in birthdays (i.e., the two husbands have the same birthday, so do the two wives, and so do the two children)?}

Answer 1

There are n families.
Let X be an R.V. denoting the number of families with same birthdays.

Here,

In the following table, we iterate n for different values and check for which smallest n, the probability is greater than 0.5.

n	p	>.5
3	2.545E-05	False
4	5.08011E-05	False
5	8.45041E-05	False
6	0.00012651	False
7	0.00017677	False
8	0.000235236	False
9	0.000301859	False
10	0.000376592	False
11	0.000459386	False
12	0.000550194	False
13	0.000648968	False
14	0.000755662	False
20	0.00155951	False
30	0.003502039	False
40	0.006159485	False
45	0.007742867	False
50	0.009489094	False
100	0.034850112	False
500	0.428154126	False
575	0.499694604	False
576	0.500609221	TRUE
577	0.501522758	TRUE

It is observed that for n=576, the required probability is first greater than 0.5.

P.S. Here, I have taken days as Monday, Tuesday, ..., Sunday(ie. 7). If Birthday means out of 365 days, replace 7 by 365 and do the same steps.

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