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Suppose, calculating the probability of having at least two peoples same birthday from 25 people. What is the sample space and outcome of the experiment? As far as I pondered, S = 365^25 and outcome 365C25

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    $\begingroup$ What do you understand sample space (and outcome) to mean? $365^{25}$ is a number - how could a number "be" a sample space? Also, is this some kind of exercise question - if so, see stats.stackexchange.com/tags/self-study/info $\endgroup$ – Juho Kokkala Oct 19 '16 at 6:18
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The sample space is the set of all possible outcomes of the experiment, corresponding to the Cartesian product of the set of $365$ possible birth dates (after hedging for pertinent caveats as to the possibility of leap years, seasonality in births, etc) with itself as many times as the number of individuals in the room.

This will produce $365^n$ ordered pairs, or $365^{25}$ $25$-tuples. From Wikipedia:

In mathematics, a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets [in this case the sample space, $\Omega$]. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

From this sample space, the event of getting two people with the same birthday can be assigned a probability. Being that we are dealing with a discrete probability space, outcomes and events can be treated as interchangeable.

The actual calculation is not $\binom {365}{25}$. The denominator will be the number of elements in the sample space, $365^{25}$, and the numerator will be much more straightforward to figure out if instead we calculate the complementary: the probability of having no matches, which can be calculated easily thinking that each person "occupies" a date that no other person can share:

$$\Pr(\text{at least 2 people share birthday})=1- \frac{365\times364\times\cdots\times \overbrace{341}^{365-n+1}}{365^{25}}$$

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