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Math/gambling wiz help needed?

Thread Starter Math/gambling wiz help needed?
HigherPOV
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Okay, I got a gambling related math/probability problem. Here it is.

If one is using a standard deck of cards and using blackjack valuations (all face cards are 10 but aces are always 1) What would be a proper O/U total for a hold-em flop so the over and under is an even money bet. If you could also provide a proof of it that would be great. My buddies and I at the poker table want another prop to bet on pre-flop.

Bets That Profit
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6.5..... King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2, Ace.... 13\2= 6.5

or add all the face value... king, jack, queen, 10.... 10x4x4(suits) = 160... ace(4x1)... 2(2x4) and so on... will equal up to 340... then 340 divided by 52 cards.. The total comes to 6.5...

So the o/u would be 6.5

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HigherPOV
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Hmmm, i don't follow. The flop has three cards. or do you mean 6.5 for each card???  I want to know the O/U for the sum total of all three cards in the flop. eg A-A-A would be 3 (lowest possible). And KKQ would be 30 (highest possible). I think the number would be between 20-30. Thanks again

Toofdoc
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My uneducated gut instinct would be 19.5

Average value of a card 6.5 with 3 per flop but now I'm questioning this?  

28 cards above and only 24 below the average.....................I'm not sure

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HigherPOV
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toof- The math tells me 19.5 also, but there are 4 face value cards of 10. Maybe just seeing so many possible 10s is ruining my perception. If it is at 19.5, most people I think would always bet the over. Because 2 face cards is a guaranteed win. Or maybe I'm missing something and 19.5 is the number. I thought it would be around 20-21.

HigherPOV
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okay, got help on math tutoring site for confirmation/clarification. this is what he said.

You can get a rough estimate by finding first the average point total of any one card and multiplying it by 3: The total number of points in the deck is

4(1+2+3+....+9+10+10+10+10)=340, so the average per card is 340/52=6.54.

For three cards drawn from three complete decks, you would have on average 3(6.54)=19.6 points.

However, your three cards come from the same deck, and this where it gets tricky. Drawing the first card will effect the average points of the remaining 51 cards: for example, if you drew a king (of value 10) first, then the average point number for the second card will be lower than 6.54 (it would be (340-10)/51=6.47).

To find the exact answer, you would have to go through all possibilities ( (king, 7, 8), (ace,5,10), etc.) and calculate the value, then find the average. I suspect it would still be close to 19.6.

Looks like O/U 19.5 is about right. So I'm going with that. seems like a fair bet to me. Thanks everyone.

Toofdoc
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Make it 20 and always bet under!

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ComptrBob
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Interesting problem.

As a poster above pointed out, this is a card selection problem without replacement, i.e. once a card is out, the remaining cards will yield a different distribution of totals.

Let Pn be the probability of a sum of the 3 cards being n. Sums can range from 3 to 30.

If we want a fair bet, we must have P3 + P4 + P5 + ... + Pk be exactly 50% where k + 0.5 is the OVER/UNDER sum. Well, this does not happen, e.g 19.5 goes OVER more than it goes UNDER.

Running a Monte Carlo simulation of the problem allows us to come up with the chance of the sum going UNDER 19.5 being roughly 47.54%. So a fair bet is 19.5 OVER -110.3, UNDER +110.3.

Similarly, if we want a fair bet skewed to the UNDER, we could offer 20.5 OVER +118.4, UNDER -118.4

HigherPOV
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thanks comptrbob, nice explanation. so lets say I decided to be a banker and offered 21 and over as one side and 19 and under as the other. While I keep the number 20. What would be the house edge on that sort of bet? And would betting under 20 and over 20 be about even?

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HigherPOV

thanks comptrbob, nice explanation. so lets say I decided to be a banker and offered 21 and over as one side and 19 and under as the other. While I keep the number 20. What would be the house edge on that sort of bet? And would betting under 20 and over 20 be about even?

19 & UNDER occurs 47.54% as I said above, 20 pushes 6.68% of the time, and 21 & OVER occurs 45.78%. So house edge Is 6.68%. You can see UNDER 20 is the small favorite vs the OVER.

Also the average sum is around 19.615.

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