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# Follow-up to the four-card flush flaw

13 December 2008

The odds of hitting *any* kind of flush from four cards should be 9/47 or 5.222 to 1.

Out of all 714 four-card flush combinations (for one suit), 36 are straight flush draws and 4 are royal draws. The odds of hitting the straight flush with a straight flush draw is 47 to 1 for insides and 23.5 to 1 for outsides. Of the 40 combinations, 8 are inside. The other 32 require one card. So, the total odds of hitting the straight flush is ((32 + 2 x 8) / (47 x 40)) or 2.5532% when you have the four card draw for a royal or straight flush up. All in all when you do the math, this reduces the odds of hitting a flush from 5.222 to 1 to 5.262 to 1.

Of all initial poker hands, the odds of getting four of one suit is (13/52 x 12/51 x 11/50 x 10/49 x 39/48) x 4 = 0.8583% or 1 out of 116.5 hands. Assuming 10 plays per minute and 100 hours of record keeping, that gives 60,000 hands. On average, he would have 515 hands that would be a four-card flush for which he would keep records.

According to the binomial distribution, the odds of getting a flush out of a 4 card is 9 / 47. At 515 trials, he should get a flush 80 times. However, the normal distribution for 515 trials with a probability of (9/47) gives a sigma of +/- 9 flushes. At 2.576 sigmas, he would be in the less than 1% range. So, unless he's getting less than 57 flushes (11%), he's just unlucky, not being cheated.

Thanks for the doing the math.

A few days ago, I posted an e-mail from Gene, who thought there might be a flaw in video poker machines because he was completing fewer four-card flush draws than he thought he should have been completing.

Gene didn't give the number of four-card flush draws he recorded or the number of times he completed the flush, so we don't know how far away from the expected number he was.

We always have to remember that the results on video poker and slot machines are determined at random. When events don't happen with the frequency predicted by the math, it doesn't automatically mean that we are being cheated. It's much more likely that we're experiencing the randomness of the machines.

Best of luck in and out of the casinos,
John

Send your slot and video poker questions to John Robison, Slot Expert, at slotexpert@comcast.net. Because of the volume of mail I receive, I regret that I can't reply to every question.

John Robison

John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots
John Robison
John Robison is an expert on slot machines and how to play them. John is a slot and video poker columnist and has written for many of gaming’s leading publications. He holds a master's degree in computer science from the prestigious Stevens Institute of Technology.

You may hear John give his slot and video poker tips live on The Good Times Show, hosted by Rudi Schiffer and Mike Schiffer, which is broadcast from Memphis on KXIQ 1180AM Friday afternoon from from 2PM to 5PM Central Time. John is on the show from 4:30 to 5. You can listen to archives of the show on the web anytime.

#### Books by John Robison:

The Slot Expert's Guide to Playing Slots