STEP II 2007 question 13 solution

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is much smaller than 1 so we can assume and higher powers are negligible.

The required probability is the same as 1-P(nobody shares a birthday). Which is where we have used .

Want the smallest natural number k such that

Notice that 22*23=506 = 2*253. So putting k=23 we have

using the approximation for log(2) given.

Also, since is strictly increasing for k in the natural numbers, k=23 is the smallest satisfying the inequality.

Suppose there are k guests. P(somebody having the host's birthday) = 1 - P(nobody has it) which is

For an approximation of k we solve

For k=253 we have equality. By a similar argument to the previous part, this is the smallest natural number satisfying this inequality.

Solution by SsEe.