"The Therory of Poker"

[high007]

hey everyone, I have been studing "The Theroy of Poker" and on page 49 Sklansky says that in a holdem game with a four flush with 2 cards to come that the odds of making the flush are 1and3/4 to 1. how did he calculate this??

Thanks

[IceKing]

Hi!

There is 5 cards that we know, that leaves 47 unknown. 9 cards out of 47 will make your flush. Two cards to come, and you will need atleast one of your suit. So we will count that how often neither of those cards is your suit. 38/47 x 37/46 = 0,65. Which means that about 1/3 of the time you will make your flush. So odds are aprox. 2 (1 and 3/4) against 1. [img]/images/graemlins/wink.gif[/img]

[daryn]

just find the odds of not making a flush, and subtract from 1.

you have a 4 flush. you have seen 5 cards, your 2 and the flop. there are 45 unseen cards. of those 45, 9 will make your flush. what are the chances you don't make it?

well, the chances of not making it on the turn are 36/45. then, then chances of not making it on the river are 35/44. multiply those 2, and the chances of not making your flush are: .63636363...

so the odds of making the flush are 1-.636363...

odds of making flush: .363636...

to convert that to x-1 against form, you just take the inverse, [1/.362626], and subtract 1, that is your x.

1.75-1 or 1 and 3/4 to 1

[IceKing]

There is 47 unkonown cards.. 52 - (2hold+3cardflop) [img]/images/graemlins/grin.gif[/img]

[daryn]

correct

funny though, that error gave me 1.75-1 exactly.

[Stallion714]

There is 5 cards that we know, that leaves 47 unknown. 9 cards out of 47 will make your flush. Two cards to come, and you will need atleast one of your suit. So we will count that how often neither of those cards is your suit. 38/47 x 37/46 = 0,65. Which means that about 1/3 of the time you will make your flush. So odds are aprox. 2 (1 and 3/4) against 1.
I dont doubt Skalansky, but your method is FLAWED. It's like saying if you need to flip heads at least one out of two tosses your odds are only 25%. Odds of flipping heads is 1/2 * 1/2 for the next flip is 1/4. So you have a 25% chance, or 1 in 4, to flip heads in two seperate random chance events? Ridic.
I have a hard time with fractions. There's almost a 40% chance that one of the next two cards is a flush card. About 20% on the turn + about 20% chance on the river. Which is 40% (or 2 times in 5 chances)
It's really 38.5% or so. A 1 in 3 chance is 1.5 to one, and a 1.75 to 1 (35%)is really as close as you can get to 38% without overestimating, and without breaking it into 1/8's or smaller. Hope it helps. It helped me out to figure this out good question.

[BruceZ]

[ QUOTE ]
There is 5 cards that we know, that leaves 47 unknown. 9 cards out of 47 will make your flush. Two cards to come, and you will need atleast one of your suit. So we will count that how often neither of those cards is your suit. 38/47 x 37/46 = 0,65. Which means that about 1/3 of the time you will make your flush. So odds are aprox. 2 (1 and 3/4) against 1.
I dont doubt Skalansky, but your method is FLAWED.

[/ QUOTE ]

His method and answer are completely correct. Actually it comes out to 35% or 1.86-to-1.


[ QUOTE ]
It's like saying if you need to flip heads at least one out of two tosses your odds are only 25%. Odds of flipping heads is 1/2 * 1/2 for the next flip is 1/4. So you have a 25% chance, or 1 in 4, to flip heads in two seperate random chance events? Ridic.

[/ QUOTE ]

You didn't go far enough. That would be 25% chance NOT to have heads on the next 2 flips, or 75% chance that you will have heads on at least 1 of the 2 flips. This is correct since there is a 25% chance of getting 2 tails. Note that you can also get this by 50% + (50% of 50%) = 50% + 25% = 75%. That is, there is a 50% chance of getting heads on the first flip, and a 50% * 50% = 25% chance of getting tails on the first flip AND heads on the second flip.

[ QUOTE ]
I have a hard time with fractions. There's almost a 40% chance that one of the next two cards is a flush card. About 20% on the turn + about 20% chance on the river. Which is 40% (or 2 times in 5 chances)

[/ QUOTE ]

You can't add these this way since the 20% on the river assumes that you missed on the turn which only happens 80% of the time. You would have to do 20% + (80% of 20%) = 20% + 16% = 36%. The actual answer is 35% since instead of 20% there is really a 9/47 = 19.1% chance on the turn and 9/46 = 19.6% chance of the river if you miss on the turn. 19.1% + (19.6% of 81.9%) =35%.

[Kurn, son of Mogh]

9/47 + 9/46 - (9/47 * 9/46)

which should equal 1.91:1

[BruceZ]

[ QUOTE ]
9/47 + 9/46 - (9/47 * 8/46)

which should equal 1.91:1

[/ QUOTE ]

That's incorrect. It should be 9/47 + 9/47 - (9/47 * 8/46) = 1.86:1. The probability of the turn card being a flush card and the probability of the river card being a flush card are each 9/47 before either of these cards are dealt. Note these two other equivalent forms:

9/47 + (38/47)*(9/46) = 1.86:1

1 - (38/47)*(37/46) = 1.86:1

[Kurn, son of Mogh]

Thank you. Although I would never even consider disputing you on probabilities, I will say that all you need to know to make the right decision at the poker tble is that it's "about 19:10."