Coin Flips......

[PAUL-IN]

can someone explain why 22 is a coin flip vs AK? (well it's basically an 11:10 ) there are five chances that AK will hit an ace or king, of which there are only six. please explain??

[Bozeman]

Approximately: 1/8*5 chance of hitting A or K=5/8, but pp will hit trips ~1/6 of these times, so 5*5/(8*6)=25/48.

[PAUL-IN]

thanks. excellent.

[DeadRed]

Be careful with some of these approximations, especially when you end up mutiplying them together, small differences can become magnified. The probability that the first card on the board doesn't pair A or K is 1/8 (6 ways to hit / 48 unseen cards), but as you remove cards that do not pair, each successive card becomes slightly more likely to hit the AK. It doesn't seem to make much difference that the fifth card on the board has a ~13.6% chance to pair instead of a 12.5% chance (assuming that the first 4 cards missed), but when you take into account the slight differences for cards 2-5, see what can happen.

With AK and 22 in the pockets, the chances that AK gets no help (another A or K) is closer to 50%, not 5/8 (62.5%). My calculation is based on the fact that there are 48 outstanding card and 42 of them are not A or K, so moving down the board, the probability that the first doesn't pair is 42/48, for the second you now have a 41/47 chance of not pairing, etc. To find the probability that all 5 cards miss the AK, take (42/48)*(41/47)*(40/46)*(39/45)*(38/44) and you get a ~49.7% chance of AK not getting paired.

The calculation for getting 1 A or K, but one (or more) 2s has a similar problem. The actual chance of this is ~13.8% and not ~16.7%. Since both miscalculations were off by about 20% of the correct values they washed each other out and you got close to the 50% value you expected.

If you take the 49.7% of the time AK gets no pair, add in the 7% of the time that AK gets one pair but a 2 comes as well (from .503 AK gets paired once times .138 a 2 makes trips), subtract off the 3.7% of the time that AK gets a straight, you get 53%/47% 22 over AK. I have left out the rest because my head is starting to swim...

* This does not take into account the very rare situations where AK makes trips, a straight, a flush, or a boat, and 22 makes a boat or quads to stay in front, but these are some tiny numbers.