Question

Assume that all (525)poker hands are equally likely. What is the probability that you will be dealt:

(a) A flush (a flush is a hand where all 5 cards are of the same suit)

Answer 1

There is 1.98% of probability of being dealt a flush in 5-card Poker

Explanation:

To know the probability of a flush being dealt, we can calculate the number of cases when that happens and divide it by the total number of cases of poker hands that exist, naming A the event of a flush.

We will use combinations (nCr button on a calculator) to count the number of cases, because we don't care about the order (it is the same to be dealt a 2, 4, 6, 7 and 8 of hearts than the opposite order), being a flush the event when we take 5 cards out of 13 of the same suit, times 1 out of 4 possible suits and the total number of cases is taking 5 random cards out of 52.

`P_(A) =(Cases   Of Flush)/(Hands) =\frac{\left(\begin{array}{ccc}13\\5\end{array}\right)*\left(\begin{array}{ccc}4\\1\end{array}\right)}{\left(\begin{array}{ccc}52\\5\end{array}\right)} =(5148)/(2598960)=0,00198\\`

That means there is about a 2% of probability of being dealt a flush.

In other words, of every 16660 plays, 33 will be, on average, a flush