Probability of a Royal Flush
The poker hand known as a royal flush consists of an ace, king, queen, jack and ten card and all must be in the same suit. The chance of the first card being an ace is 4/52 since there are 4 ace cards in a deck of 52 cards and we don't care about suit. The probability that we'll next draw a king of the same suit is now 1/51 since there is only one king of the same suit in the remaining 51 cards in the deck.
To calculate the highly unprobable scenario of drawing a royal flush in order from a 52 card deck:
4/52 * 1/51 * 1/50 * 1/49 * 1/48 = 4 / 311,875,200
So the odds are 1 in 78 million.
Of course the odds are much better that you will draw a royal flush in any order in a poker game. Now, the odds of the first card is 20/52 because we have five different suits to draw (spade, club, heart, and diamonds) from four cards (ace, king, queen, or jack). Now we must take into account all the possible permutations (order) the cards may be drawn which are acekingqueenjackten, tenjackqueenkingace, acejacktenkingqueen, etc.
The permutations are calculated by a five factorial: *5 * 4 * 3 * 2 * 1 = 120
20/52 * 1/51 * 1/50 * 1/49 * 1/48 * 120 = 2400 / 311,875,200
So the odds for any royal flush being drawn are one in 130,000!
