# Is $5$ hits the same to $15$ hits?

In a universe of $25$ numbers. At the time of the drawing you take $15$ numbers (non-repeatable)

I pick $15$ numbers.

The chances of having $15$ hits is the same to have exactly $5$ ?

Example:

Universe: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25
My picks are: 1,2,3,4,5,6,7,8,9,10,13,14,15

Drawing: 1,5,6,7,8,10,12,13,14,16,18,20,22,24,25

What are the odds of hit $15$ numbers What are the odds of hit $5$ numbers (but only $5$, not more)

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You pick 15 numbers and the chances of having 15 hits....what's a hit here? –  DonAntonio Aug 16 '13 at 14:40
sorry my native is not english. A hit is having the number picked in the drawing. In the case of the example, I picked number 1 and the draw have number 1 so I have a hit there. –  Bart Aug 16 '13 at 14:41

There are $\binom{25}{15}$ possible draws, only one of which gives you $15$ hits, so the probability of getting $15$ hits is the very small number

$$\frac1{\binom{25}{15}}=\frac1{3,268,760}\approx0.000000306\;.$$

A draw that gives you exactly $5$ hits must contain $5$ of your $15$ numbers and all $10$ of the numbers that you did not choose; there are $\binom{15}5$ such sets, one for each set of $5$ of your choices. The probability that you will get exactly $5$ hits is therefore

$$\frac{\binom{15}5}{\binom{25}{15}}=\frac{3003}{3,268,760}\approx0.000918697\;.$$

This is still small, but it’s much bigger than your chance of getting $15$ hits.

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Thank you very much! –  Bart Aug 16 '13 at 14:43
@Bart: You’re very welcome. –  Brian M. Scott Aug 16 '13 at 14:44