Comparing the Odds of a Royal Flush in Poker to Winning the Mega Millions Lottery Jackpot

To obtain a royal flush of spades in poker, using just five cards and with nothing wild, you need to pull five exact cards from a stack of 52 cards. The odds of this are so long that its rarely played this way among peers or at the (see for some online casinos).

To pull the first card, you have a chance of pulling one of the five cards from the 52, a 5-in-52, or 9.2% chance.
The odds of pulling a second card is four in 51, because you only want one of four possible remaining cards. a 4-in-51 chance, or 7.8% chance.

To calculate the odds of both of these two steps happening, you must multiply them together. One easy way is to multiply the 5 and the 4 (20) and multiply the 52 and 51 (2652), and divide the small one by the large one: 20/2652 = 1/132 or 0.7%.

If we pull another card, looking for one of the three remaining in our precious royal flush, our odds are 3/50.
For the fourth, its 2/49.
For the fifth, to get the final card, its a 1/48 chance.

If you multiply these all together, you get (5/52)*(4/51)*(3/50)*(2/49)*(1/48) = 120/ 311,875,200 or 1 in 2,598,960. PokerSites can guide you to a poker website. Therefore, at home or at a casino, you have a one in two and a half million chance of pulling a royal flush of spades in five cards from a deck of 52 cards.

Away from poker sites for a moment, consider the lottery:

Playing the California lottery, the largest prizes are awarded in the "Mega Millions" game.

The Mega Millions game is played by choosing six numbers, including a "Mega Number" from a group of 56 possible numbers. Actually, the "Mega Number" can only be between 1 and 46 and is drawn from a seperate set of 46 numbers.

I was surprised at how closely it compares to pulling the royal flush of spades from a deck of cards.

The winning numbers are chosen by some kind of rubber ball ceremony which takes place in Atlanta, Georgia. The odds of the first ball corresponding to one on your ticket is 5 in 56, an 8.9% chance. The odds of the second ball corresponding to one of your numbers is 4 in 55, a 7.2% chance.

Again, to calculate the odds of both of these two steps happening, you must multiply them together. 5/56 x 4/55 = 20/3080 or 1 in 154 or 0.6%.

The odds of getting the third number are 3/54.
For the fourth, it is 2/53.
For the fifth, it is 1/52.


If you only had to get 5 numbers right, the odds of winning Mega Millions would be (5/56) x (4/55) x (3/54) x (2/53) x (1/52), or 120/458,377,920 or 1 in 3,819,816.

This far in the calculations, it's almost exactly like trying to get a royal flush of spades out of a 56-card deck. However, the Mega Millions game also has the Mega Number, a 1 in 46 chance pulled from a seperate array of balls in Atlanta. Having to also successfully select this number decreases your odds of winning by 46 times.

Requiring a Mega Number changes the odds from 1 in 3 million to 1 in 175 million (175,711,536).

So, when you see someone get a royal flush of spades in a poker game at a casino in a movie, remember how outrageously long those odds are. Its about as hard as hitting all five numbers in the lottery.

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