Video poker -- odds of getting quads

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.

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

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.

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"??

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 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.

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.

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.

The probability of getting quads within 420 hands is actually instead:

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.

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.

One more question and I just want to be sure about this: when you say "probability" and when you say "the expected number" are they the same thing as saying "odds of getting"?

And just to repeat what I wrote above: Is the phrase "your chance of a royal flush is 1 in 40,000 on this hand, and the next hand, and the next ten thousand hands" actually incorrect?

Quote: AlanMendelson

One more question and I just want to be sure about this: when you say "probability" and when you say "the expected number" are they the same thing as saying "odds of getting"?

And just to repeat what I wrote above: Is the phrase "your chance of a royal flush is 1 in 40,000 on this hand, and the next hand, and the next ten thousand hands" actually incorrect?

No. The probability at least how I am using it in this case is for the event to happen "at least once". "The Expected Number" is the average number of these events occurring.

So at least one Royal flush happening within 40,000 hands has a probability of 0.6321.

The expected number of Royal flushes within 40,000 hands is one.

The Royal flush breakdown over 40,000 hands looks like this: Probabilities and Royals aren't exactly one since it is possible to get 11+ Royals in 40,000 hands and I didn't want the table to go on forever.

Royals	Probability	Expected Royals
0	0.3678748426	0
1	0.3678840397	0.3678840397
2	0.1839420199	0.3678840397
3	0.0613124737	0.1839374212
4	0.015327352	0.061309408
5	0.0030652405	0.0153262024
6	0.0005108223	0.003064934
7	7.29654958949144E-005	0.0005107585
8	9.11931884961284E-006	7.29545507969027E-005
9	1.01308032543513E-006	9.11772292891614E-006
10	1.01287770430451E-007	1.01287770430451E-006
10 Royals or less	0.99999999	0.9999998887

The correct phrase is: The odds of getting a Royal Flush on the very next hand is always 1 in 40,000. (approximately)

For any stretch of multiple hands, you need to do the math.

The formula for the math is: 1 - (39999/40000)^N
where N = the number of hands played.

Posting consolidation.

But the answer to the original question is NO just because you don't get quads this hand, your odds of getting quads on the next hand are not any different. Independently, each hand has the same odds as the last whether you got quads or not. Quads are not "due".

ZCore13

The probability of 4oak on ONE trial = 1 in 420
lets call that P = 1/420

edit:(damn head cold)
The probability of hitting the very next 4oak on the 10th trial would be not hitting it on the first 9 and hitting it on the 10th
(1-P)^9 * P
The probability of hitting the very next 4oak on the 9th trial would be not hitting it on the first 8 and hitting it on the 9th
(1-P)^8 * P
work you way to the first trial (these are the relative probabilities)
add these all up and
Exactly what the Wizard computed. 2.3556%
(is not 1/42 but that is close)

This is the probability of getting at least 1 4oak in the first 10 trials.
You could get more than 1.
(more math required to get those probabilities)

remember P always equals 1/420 for any ONE trial
P = 1/420
P never changed

Now the Degree of Certainty (DC)
for an independent event certainly increases for
at least one success as the number of TRIALS also increases.
This should be very intuitive.

The 50/50 mark in trials to get at least 1 4oak at P = 1/420 for each trial
=LOG(1-(0.50))/LOG(1-(P))
This = 290.78 Trials

So in your very next 291 hands played it is a coin flip (a 50/50 shot) to get at least 1 4oak

965.93 trials would be for the Degree of Certainty of 90.0% of getting at least 1 4oak.
of course that means a 10% chance (a high relative value) of NOT getting at least 1 4oak.

Now, one still has to look at the probability as before the very first trial begins.
It should also be intuitive that if you did NOT get a 4oak on the first 100 trials,
you do NOT still have a 50/50 shot of hitting a 4oak.
You only have 191 trials left to get at least 1, NOT 291 trials.
That probability has decreased to 36.5745% of getting at least 1 4oak in the next 191 trials

it is true that P never changes for a 4oak for ONE hand
but as the number of TRIALS increase the Degree of Certainty of getting at least 1 4oak
increases, and it has to. it is a certainty. From the start of the # of trials.

Do not confuse, as many do,
the Probability of an event for ONE trial VS. the same probability of an event over MANY trials.

It is just simple math and not a Gambler's Fallacy unless you think that at
trial #291 you still have a 50/50 shot at hitting a 4oak because that was the same before you started the 291 trials.
This is part of the Gambler's Fallacy.

There are many things dealing with the probability of an independent event
that is not very intuitive, almost at times sounds wrong and therefore must be wrong.

But by remembering the basics of probability and using them, the truth will always be in the math.

Quote: AlanMendelson

So does this summarize it:

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.

The last statement is incorrect.

Think about a single die. The odds of rolling a 6 is 1 in 6 (or 1 in 6 rolls).

The odds of rolling a 6 in the next 6 rolls in not 1 out of 1. Nor is the odds of rolling a 6 in the next 3 rolls 1 in 2.

You have to look at the odds of something NOT happening at all when there is chance something may occur more than once in a given period.