Question

Calculate the number of possible five-card poker hands, dealt from a deck of 52 cards. (The order of cards in a hand does not matter.) A royal flush consists of the five highest-ranking cards (ace, king, queen, jack, 10) of any one of the four suits. What is the probability of being dealt a royal flush (on the first deal)?

Short answer

The chances of getting a royal flush on the first deal are slim $P_{\text{flushes}}=1.539\times {10}^{-6}$

Step-by-step solution

Step1:definition of probability

Probability is a measure of the likelihood of an event occurring. Many events are impossible to predict with 100% accuracy. We can only predict the likelihood of an event occurring using it, that is, how likely it is to occur. Probability can range between zero and one, with zero indicating an impossible event and one indicating a certain event.

Step2:Probability of royal flushes

Because there are$52$ cards in the deck, the number of ways to choose five cards from this deck is:

$\mathrm{\Omega }(52,5)=\frac{52!}{5!(52-5)!}=2598960$

There are four royal flushes among these ways, so the probability of these flushes is:

$P_{\text{flushes}}=\frac{4}{2598960}=1.539\times {10}^{-6}$

$P_{\text{flushes}}=1.539\times {10}^{-6}$