# Odds Of Picking Five Specific Cards From A Pool Of 20
**URL:** https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854
**Category:** Factual Questions
**Created:** [June 21, 2019, 5:14pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854 "2019-06-21T17:14:58Z")
**Posts on this page:** 16
**Page:** 1
### Author:  [@HeyHomie](https://boards.straightdope.com/u/HeyHomie)
#### Post date: [June 21, 2019, 5:14pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/1 "2019-06-21T17:14:58Z")
An online poker game I play has this side game that players can qualify for, every few dozen or so hands. You’re given five cards needed to make a Royal Flush (10-J-Q-K-A, in one suit) and 15 random cards. Get the Royal Flush, win tens of millions of in-game chips.
What are the odds?
---
### Author:  [@zimaane](https://boards.straightdope.com/u/zimaane)
#### Post date: [June 21, 2019, 5:25pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/2 "2019-06-21T17:25:25Z")
In order to win you need to get a winning card each draw:
1st draw: 5/20
2nd draw 4/19
3rd draw 3/18
4th draw 2/17
5th draw 1/16
Multiplying those fractions gives 120/1860480, or 0.000645.
Good luck!
---
### Author:  [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)
#### Post date: [June 21, 2019, 5:54pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/3 "2019-06-21T17:54:30Z")
What **zimaane** said.
The fractional version simplifies to 1 chance in 15,504, which may be the easiest way of thinking about it.
Or if you get 10 chances a day to play this game, you should win once every 4 years and 3 months, on average.
---
### Author:  [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)
#### Post date: [June 21, 2019, 6:16pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/4 "2019-06-21T18:16:59Z")
> [@zimaane](#):
>
> In order to win you need to get a winning card each draw:
>
> 1st draw: 5/20
> 2nd draw 4/19
> 3rd draw 3/18
> 4th draw 2/17
> 5th draw 1/16
>
> Multiplying those fractions gives 120/1860480, or 0.000645.
>
> Good luck!
I think you’re missing a zero—0.0000645.
---
### Author:  [@zimaane](https://boards.straightdope.com/u/zimaane)
#### Post date: [June 21, 2019, 6:28pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/5 "2019-06-21T18:28:32Z")
> [@Half\_Man\_Half\_Wit](#):
>
> I think you’re missing a zero—0.0000645.
You are correct. My bad - hope no one got their hopes too high.
---
### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [June 21, 2019, 6:29pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/6 "2019-06-21T18:29:58Z")
Strictly speaking, the chances are _slightly_ better, because it’s possible that the 15 random cards contain another royal flush in a different suit.
---
### Author:  [@zimaane](https://boards.straightdope.com/u/zimaane)
#### Post date: [June 21, 2019, 6:48pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/7 "2019-06-21T18:48:36Z")
> [@Chronos](#):
>
> Strictly speaking, the chances are _slightly_ better, because it’s possible that the 15 random cards contain another royal flush in a different suit.
I interpreted is as 15 random cards between 2 and 9 that could never be part of a royal flush. I could be wrong though.
---
### Author:  [@Quercus](https://boards.straightdope.com/u/Quercus)
#### Post date: [June 21, 2019, 7:20pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/8 "2019-06-21T19:20:51Z")
> [@Chronos](#):
>
> Strictly speaking, the chances are _slightly_ better, because it’s possible that the 15 random cards contain another royal flush in a different suit.
Given that it’s an online poker add-on game, I’d strongly bet, either:
a) the other 15 cards are chosen to exclude the possibility of a royal flush, or
more likely, b) the programmer didn’t even think of that possibility and so the program doesn’t check for that. You could show 10-A of a different suit and the program would say “Sorry! Better luck next time”
---
### Author:  [@HeyHomie](https://boards.straightdope.com/u/HeyHomie)
#### Post date: [June 21, 2019, 8:36pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/9 "2019-06-21T20:36:32Z")
On the distribution of the non-essential cards:
I don’t know the algorithm, but I do know that the cards are such that every possible winning hand (pair, two pair, etc.) can come up. I’ve seen face cards of other suits come up as well, but I don’t think I’ve seen enough so that it’s possible to get a royal flush in either of two suits. But the algorithm does make sure that it’s possible to get a royal flush in one suit.
---
### Author:  [@Sage\_Rat](https://boards.straightdope.com/u/Sage_Rat)
#### Post date: [June 21, 2019, 9:16pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/10 "2019-06-21T21:16:42Z")
> [@Quercus](#):
>
> Given that it’s an online poker add-on game, I’d strongly bet, either:
> a) the other 15 cards are chosen to exclude the possibility of a royal flush, or
>
> more likely, b) the programmer didn’t even think of that possibility and so the program doesn’t check for that. You could show 10-A of a different suit and the program would say “Sorry! Better luck next time”
Obviously, there’s no telling how well this is all coded up and, if real money is on the line, I would expect that the servers are outside of the US and that you likely can’t sue them if the implementation doesn’t follow a reasonable reading of the stated rules.
But, I would doubt that an online casino would be stupid about probabilities.
If they’re giving tens of millions of credits for a 1 in 15.5k probability in the worst case then either credits aren’t worth much money or they’re more interested in making sure that there are enough jackpots going on to entice people into playing their other games, that they don’t mind losing out the tens of millions. Since you can only play the game after playing a dozen other games, the frequency is pretty strongly throttled. They can lose at a slow enough rate that their wins far exceed it, even with what are relatively good odds for the players in terms of a payout/probability ratio.
---
### Author:  [@Helmut\_Doork](https://boards.straightdope.com/u/Helmut_Doork)
#### Post date: [June 21, 2019, 9:25pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/11 "2019-06-21T21:25:43Z")
> [@zimaane](#):
>
> In order to win you need to get a winning card each draw:
>
> 1st draw: 5/20
> 2nd draw 4/19
> 3rd draw 3/18
> 4th draw 2/17
> 5th draw 1/16
>
> Multiplying those fractions gives 120/1860480, or 0.000645.
>
> Good luck!
sorry for the hijack, would this be the same method to figure the odds in Keno of picking exactly 20 drawn numbers correctly of 80 possible numbers, as 5/20 = 20/80? Something so rare I don’t think it has ever happened, but has a payout of not even a million?
---
### Author:  [@DPRK](https://boards.straightdope.com/u/DPRK)
#### Post date: [June 21, 2019, 9:41pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/12 "2019-06-21T21:41:14Z")
> [@Helmut\_Doork](#):
>
> sorry for the hijack, would this be the same method to figure the odds in Keno of picking exactly 20 drawn numbers correctly of 80 possible numbers, as 5/20 = 20/80? Something so rare I don’t think it has ever happened, but has a payout of not even a million?
Same “method”, sure, but picking 20 out of 80 gives astronomical odds compared to 5 out of 20.
ETA since the Keno odds are so long, I would use a different method involving approximations instead of trying to compute the exact value. Or just feed it into the computer.
---
### Author:  [@naita](https://boards.straightdope.com/u/naita)
#### Post date: [June 21, 2019, 9:44pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/13 "2019-06-21T21:44:31Z")
> [@Helmut\_Doork](#):
>
> sorry for the hijack, would this be the same method to figure the odds in Keno of picking exactly 20 drawn numbers correctly of 80 possible numbers, as 5/20 = 20/80? Something so rare I don’t think it has ever happened, but has a payout of not even a million?
Same method, and the odds are 1 in 3,535,316,142,212,174,320.
---
### Author:  [@Helmut\_Doork](https://boards.straightdope.com/u/Helmut_Doork)
#### Post date: [June 21, 2019, 9:45pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/14 "2019-06-21T21:45:55Z")
> [@DPRK](#):
>
> Same “method”, sure, but picking 20 out of 80 gives astronomical odds compared to 5 out of 20.
thanks, yes the odds are over a quintillion, but wasn’t sure of the method.
If I can continue the hijack once more, why is there a point when Keno odds do this:
1 1 in 46.446
2 1 in 20.115
3 1 in 8.009
4 1 in 4.877
5 1 in 4.287
6 1 in 5.258
7 1 in 8.826
8 1 in 20.055
9 1 in 61.420
More likely to get 5 of 20 than 4 or 20, if I understand correctly?
---
### Author:  [@DPRK](https://boards.straightdope.com/u/DPRK)
#### Post date: [June 21, 2019, 10:14pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/15 "2019-06-21T22:14:26Z")
> [@Helmut\_Doork](#):
>
> thanks, yes the odds are over a quintillion, but wasn’t sure of the method.
>
> If I can continue the hijack once more, why is there a point when Keno odds do this:
>
> 1 1 in 46.446
> 2 1 in 20.115
> 3 1 in 8.009
> 4 1 in 4.877
> 5 1 in 4.287
> 6 1 in 5.258
> 7 1 in 8.826
> 8 1 in 20.055
> 9 1 in 61.420
>
> More likely to get 5 of 20 than 4 or 20, if I understand correctly?
Roughly speaking, each of your 20 picks has a 1/4 chance of hitting a number, so on average the most likely result is that you would expect to get 5 out of 20.
---
### Author:  [@naita](https://boards.straightdope.com/u/naita)
#### Post date: [June 21, 2019, 10:50pm UTC](https://boards.straightdope.com/t/odds-of-picking-five-specific-cards-from-a-pool-of-20/835854/16 "2019-06-21T22:50:18Z")
I was trying to do a slightly more comprehensive, but still simplified explanation to expand on what DPRK wrote, and I don’t think any approach really adds much until you get to actually learning how the hypergeometric probability distribution is calculated.
Which isn’t all that hard, but requires understanding combinations and permutations.