Analysis Request UTH 1X bet

Assuming a board with no pair and no chance for either a straight or flush. What is the correct math to determine the dealers chance of pairing?

The dealer has a 15 to 45 chance of hitting a pair on the first card, I think. If he doesn't match on the first card, he has the same 15 plus a chance of matching his first card, so 18 to 44 chance. My problem is how do you add the two chances together.

You'll see this exact scenario at https://wizardofodds.com/games/ultimate-texas-hold-em/ - also you can work it out by hand. For simplicity the scenario ignores double outs, but in theory you could analyse all 990 unique outs and just count up how many you can beat/tie/lose to.

I suspect with no pair on board and no 4-card flush or straight, you play with 2nd kicker where no flush possible or 2nd kicker and other card plays.

Quote: zrlcsx

Assuming a board with no pair and no chance for either a straight or flush. What is the correct math to determine the dealers chance of pairing?

The dealer has a 15 to 45 chance of hitting a pair on the first card, I think. If he doesn't match on the first card, he has the same 15 plus a chance of matching his first card, so 18 to 44 chance. My problem is how do you add the two chances together.

15
---
45

Plus

18
---
44

Find a common denominator. Multiply 45* 44 = 1980.
Since we multiplied the 45 times 44 we need to multiply the 15 times 44 (660)
Since we multiplied the 44 times 45 we need to multiply the 18 times 45 (810)

660 + 810 = 1470

1470 / 1980 = 74.24%

74.24% seems high. According to the Wizards it is about 60.3%. This is from his UTH calculator, but I don't know how he comes up with that number. I'm not sure adding them together like straight math is the correct. I think that will result in counting the times when both cards match twice, overstating the dealers chances.

Quote: BTLWI

15/45 Plus 18/44

Ignoring your cards, the chances of hitting with first card is 15/45 = .333333. However the chances of hitting a pair with the second card is (not having hit with the first)*(hitting with second) = 2/3*18/44. This gives .606.

I suspect your cards affect this, since you're known not to have any of the 15 matches. On the second draw there are 2 chances out of 8 that the dealer's card matches yours, thus on the second turn there are (18 18 18 18 18 18 17 17) outs.

P1 = 15/43 (as you have two non-outs).
P2 = 28/43 * (142/8) / 42 (average number of outs is 18+18...+17)/8)
Total = 62.4%

btw an easier way is to work out how to miss twice = 28/43 * 24.25/42= 37.6% and then take it from 1. This is a comment technique for problems like how likely are you to throw a six in four rolls of a die = 1 - (5/6) ^ 4.

Quote: charliepatrick

Ignoring your cards, the chances of hitting with first card is 15/45 = .333333. However the chances of hitting a pair with the second card is (not having hit with the first)*(hitting with second) = 2/3*18/44. This gives .606.

I suspect your cards affect this, since you're known not to have any of the 15 matches. On the second draw there are 2 chances out of 8 that the dealer's card matches yours, thus on the second turn there are (18 18 18 18 18 18 17 17) outs.

P1 = 15/43 (as you have two non-outs).
P2 = 28/43 * (142/8) / 42 (average number of outs is 18+18...+17)/8)
Total = 62.4%

btw an easier way is to work out how to miss twice = 28/43 * 24.25/42= 37.6% and then take it from 1. This is a comment technique for problems like how likely are you to throw a six in four rolls of a die = 1 - (5/6) ^ 4.

Thank you charlie.

If I understand correctly the best way to analyse this is to calculate a miss-miss and subtract from 1. This give a result closer to the Wizard, but still not exactly. I agree my cards will affect the dealer's chances but only on his second card. For the first card there are still 45 cards in the deck, 15 of which are an out. If he happens to match one of mine on the first draw it would reduce his outs by 1 unless i happened to have a pocket pair.

From the Wizard's calculator: (chance of pairing/qualifying))

Non pair in my hand - 60.3%
Pair in my hand - 60.4%

Ignoring my hand - 60.6% 1 - (30/45)*(26/44)

Now this makes sense to me. I would really like to hear the Wizards comments.