AlanMendelson
Joined: Oct 5, 2011
• Posts: 3126
April 13th, 2013 at 9:36:34 AM permalink
Guys, please help me out. I am math illiterate. I know that the odds of getting quads in a hand of video poker are approximately 1/420. I know that the odds of getting a royal flush are approximately 1/40,000.

My question is this: do the odds change with an increased number of hands played?

This question came up on my forum where someone (who I understand is banned from this site) says that while the odds of getting quads in video poker is 1/420 on one hand, the chance of getting quads for the next ten hands is reduced to 1/42.

I always thought that:

the chance of getting a royal flush is always 1/40,000 on each and every hand
the chance of rolling a 7 with two dice is always 1/6
the chance of getting quads is always 1/420 on every hand
the chance of getting 00 on a roulette wheel is 1/38
and that previous rolls, spins, deals have no influence on future rolls, spins, deals

Specifically, here is the question:

If your chance of making quads in a hand of video poker is 1/420, what are the odds of making at least one quad in the next 10 hands?

Thanks.
Zcore13
Joined: Nov 30, 2009
• Posts: 2464
April 13th, 2013 at 10:13:46 AM permalink
The person (who is probably banned from this site for making such stupid accusations as this one) is incorrect. The deck re-shuffles every hand in video poker. Each independent hand has the same odds to start with. You could get no quads in 419 hands in a row and the next hand you play has a 1 in 420 chance to get them.

ZCore13
I am an employee of a Casino. All the personal opinions I post are my own and do not represent the opinions of the Casino or Tribe that I work for.
ahiromu
Joined: Jan 15, 2010
• Posts: 1857
April 13th, 2013 at 10:38:56 AM permalink
Quote: Zcore13

The person (who is probably banned from this site for making such stupid accusations as this one) is incorrect.

The probability of getting quads for EACH hand is 1/420 and nothing will change that. The probability, out of TEN hands, of seeing quads... including more than one hand... is (1-(419/420)^10) or about 1/42.45. Out of 420, it's 1/1.58. I request someone else check this math, I haven't done something like this in quite awhile.

Your person's logic works for the "expected" number of quads. So like out of 10 hands, your expected number of quads is 1/42. Out of 420 hands, you expect to see one quad.
Its - Possessive; It's - "It is" / "It has"; There - Location; Their - Possessive; They're - "They are"
AlanMendelson
Joined: Oct 5, 2011
• Posts: 3126
April 13th, 2013 at 10:51:37 AM permalink
Is there a difference between "odds of getting quads" and "expected number of quads" ??

When you say "out of 10 hands, your expected number of quads is 1/42" does that mean the same thing as "if you play ten hands the odds of getting quads is 1/42"??
Wizard
Joined: Oct 14, 2009
• Posts: 15658
April 13th, 2013 at 11:00:00 AM permalink
Quote: AlanMendelson

If your chance of making quads in a hand of video poker is 1/420, what are the odds of making at least one quad in the next 10 hands?

1-(1-(1/420))^10 = 2.36%.
It's not whether you win or lose; it's whether or not you had a good bet.
ahiromu
Joined: Jan 15, 2010
• Posts: 1857
April 13th, 2013 at 11:04:53 AM permalink
It comes down to definitions (which I might be looking at incorrectly), and "odds of getting quads" leaves out the chance of multiples when you're talking about multiple hands... which expected number does not. Also, because of this, expected number can get 1 or higher (because it CAN happen more than once) but "odds" always leaves the very real chance of never happening (like seeing 10 reds in a row).

Fictional 36 number roulette wheel, 18 and 18:
Odds of seeing "red" at least once out of 10 spins - (1-(1/2)^10) = 999 out of 1000.
Expected number of reds out of 10 spins - 5.

Edit: I think I AM looking at this incorrectly, I'll leave what I said but please take it with a grain of salt.
Its - Possessive; It's - "It is" / "It has"; There - Location; Their - Possessive; They're - "They are"
AlanMendelson
Joined: Oct 5, 2011
• Posts: 3126
April 13th, 2013 at 11:51:41 AM permalink
So does this summarize it:

The odds of getting a royal flush are one in 40,000 hands. And in 40,000 hands you should expect to get one royal flush.

---AND----

The odds of getting a royal flush are one in 40,000 hands. And in the next one thousand hands you should expect to get one-fortieth (1/40) of a royal flush.

---AND----

The odds of getting a royal flush are one in 40,000 hands. And the odds of getting a royal flush in the next one thousand hands is 1 out of 40.

Are all three of these statements correct? If so, what happened to the "statement" (for lack of a better word) that "your chance of a royal flush is 1 in 40,000 on this hand, and the next hand, and the next ten thousand hands"?

Thanks.
tringlomane
Joined: Aug 25, 2012
• Posts: 5382
April 13th, 2013 at 11:52:57 AM permalink
Quote: Zcore13

The person (who is probably banned from this site for making such stupid accusations as this one) is incorrect.

Is it stupid? I expect anyone who hasn't touched probability in over a decade to give an incorrect answer like this one. And given the small sample of hands, it's not even very far off from the correct answer.

As the Wiz and others have pointed out, Alan. You have a 1 in 420 probability to hit quads and a 419 in 420 probability to not hit quads in any given hand. And the answer is 1 - (419/420)^10 = 2.36%. 2.36% is 1 in 42.55.

The reason why we calculate it this way with respect to the exponential, (419/420)^10, is that these are independent events always with the same result. Failing to hit quads. So (419/420)^10 = 0.9764 = 97.64% is the probability you will fail to hit any quads in the next 10 hands. The probability to hit at least one quad is simply all the other possibilities and the sum of all probabilities are 1. So hitting at least one quad is 1 - 0.9764 = 0.0236 = 2.36%.

The reason why it's slightly less than 1 in 42 to see any quads is that there is also a small probability you can get 2 or more quads in 10 hands. Hopefully you're playing "Shockwave Poker" when that event happens. This effect is more clear when you do the math near the 420 hand mark. The poster on your forum would say it's 100% likely that you will get quads within 420 hands, which is definitely not the case.

1 - (419/420)^420 = 0.6326 = 63.26%

This is roughly the probability of getting a Royal within one "Royal Cycle" as well (approx. 40000 hands)
1 - (39999/40000)^40000 = 0.6321 = 63.21%.

Hope this makes things a little more clear for you and others.
tringlomane
Joined: Aug 25, 2012
• Posts: 5382
April 13th, 2013 at 11:59:30 AM permalink
The middle one is closest to correct. The first two statements to be correct would be more like this:

We expect to average one Royal after 40000 hands. We expect to average 1/40th of a Royal after 1000 hands. The third statement, see my previous post for details, but the probability of at least one Royal in the next 40k hands is 1 - (39999/40000)^1000 = 0.0247 = 2.47% = 1 in 40.50.
AlanMendelson
Joined: Oct 5, 2011