Mathematical probability of getting a low, challenge

[TheHammer24]

My brother, myself, and some friends of mine got into a big debate over probability of getting a low in poker (Defined as 5 unpaired cards under 8, A's low). Here's our series of questions some of which we found answers.

1. How many possible 5 card lows are there disregarding suits?

2. What is the chance of dealing five cards out and having them qualify as a low?

3. What is the proportion of low hands to total poker hands?

4. What is the chance of making a low if you deal out 7 cards?

For number 1 we are pretty confident the answer is 56. Number 2, my brother felt it was something along the lines of 0.02%, which is entirely too low. I think it has to be somewhere around 3%. 3 and 4 we could not figure out. Anyone know these answers and can show the mathematics behind them?

[Siegmund]

1. Yes, 56 (if you meant "8 or lower," the usual low qualifier. Only 35 if you really meant "lower than 8" like you said.)

2/3. Your brother just has a decimal point in the wrong place.
Each of those 56 lows can be made in 4^5=1024 ways if flushes don't count. So, 57344 of the 2598960 5-card poker hands qualify for low, which is 2.206% of them.

#4 requires a bit more thought. It's late at night so I'll probably enumerate these wrong, and there is probably a shortcut I am overlooking, but here goes:

You might hold 7 low cards, 6 low cards and 1 high card, or 5 low cards and 2 high cards.

If you hold 7 low cards, you might have

7 Different cards - 8 * 4^7
One pair - 8 * 6 * (7C5)*4^5
Two pair - 8C2 * 6^2 * (6C3)*4^3
Trips - 8 * 4 * 7C4 * 4^4

131072 + 1032192 + 1290240 + 286720 = 2740224

If you hold 6 low cards and one high card:

All different - 20 * 8C6 * 4^6
One pair - 20 * 8 * 6 * (7C4)*4^4

20 * (114688 + 430080 ) = 10,895,360

5 low cards and two high cards:

The 5 low cards must all be different: 56 * 4^5

The 2 high cards can be a pair: 5 * 6
or they can be two ranks: 10 * 4^2

57344 * 190 = 10,895,360

For a total of 24,530,944 low-qualifying hands
out of 133,784,560 possible 7-card hands: 18.34%.

I won't be surprised if a small sleep-induced arithmetical error has crept in, but the 15-20% ballpark matches my intuition.

And, incidentally, congratulations on asking an "obvious" poker question that has not come up on this forum recently and actually gave something new to think about!

[BruceZ]

[ QUOTE ]
1. Yes, 56 (if you meant "8 or lower," the usual low qualifier. Only 35 if you really meant "lower than 8" like you said.)

2/3. Your brother just has a decimal point in the wrong place.
Each of those 56 lows can be made in 4^5=1024 ways if flushes don't count. So, 57344 of the 2598960 5-card poker hands qualify for low, which is 2.206% of them.

#4 requires a bit more thought. It's late at night so I'll probably enumerate these wrong, and there is probably a shortcut I am overlooking, but here goes:

You might hold 7 low cards, 6 low cards and 1 high card, or 5 low cards and 2 high cards.

If you hold 7 low cards, you might have

7 Different cards - 8 * 4^7
One pair - 8 * 6 * (7C5)*4^5
Two pair - 8C2 * 6^2 * (6C3)*4^3
Trips - 8 * 4 * 7C4 * 4^4

131072 + 1032192 + 1290240 + 286720 = 2740224

If you hold 6 low cards and one high card:

All different - 20 * 8C6 * 4^6
One pair - 20 * 8 * 6 * (7C4)*4^4

20 * (114688 + 430080 ) = 10,895,360

5 low cards and two high cards:

The 5 low cards must all be different: 56 * 4^5

The 2 high cards can be a pair: 5 * 6
or they can be two ranks: 10 * 4^2

57344 * 190 = 10,895,360

For a total of 24,530,944 low-qualifying hands
out of 133,784,560 possible 7-card hands: 18.34%.

I won't be surprised if a small sleep-induced arithmetical error has crept in, but the 15-20% ballpark matches my intuition.

And, incidentally, congratulations on asking an "obvious" poker question that has not come up on this forum recently and actually gave something new to think about!

[/ QUOTE ]

I get exactly the same.