The Odds of Being Dealt Pocker Aces 4 Times in 100 hands?

[Idiotex]

Can somebody give me the answer to this along with how they came to their answer?

Cheers.

[rufus]

The probability of getting pocket aces is 1/221.
So the probability of getting exactly 4 pairs of pocket aces will be:
(1/221)^4*(220/221)^96*(100 choose 4)

This works out to roughly 1 in 1000

[Idiotex]

Thankyou.

[jesse8888]

Worth noting OP:

rufus's answer is completely correct, and he specifically does not answer the "odds of getting dealt pocket aces 4 or more times in 1000 hands". The answer to this question is a slightly larger number.

In this case, the actual difference would be very minimal, as the event in question (being dealt aces) is very rare.

However, if you wish to apply his formula to other problems, just be sure you clearly understand what it's solving for.

[kinghippo423]

I don't get the 100C4 part in the formula of rufus. What 100C4 means in that case? I'm confused a lil'.

If I want to know the odds of having one time AA in 180 hannds I would do (1/221)^1 * (220/221)^179 * ??? = ???

Thanks in advance !

[filsteal]

[ QUOTE ]
I don't get the 100C4 part in the formula of rufus. What 100C4 means in that case? I'm confused a lil'.

If I want to know the odds of having one time AA in 180 hannds I would do (1/221)^1 * (220/221)^179 * ??? = ???

Thanks in advance !

[/ QUOTE ]

100C4 is shorthand for "100 choose 4," which is defined as

100!/[4! (100-4)!].

It's the number of ways to choose the 4 hands where you got aces from the 100 total hands where it was possible (i.e., you could've gotten aces on hands 1,5,7,13, or on hands 16,24,67,93, etc.)

As for your question, the "???" on the left of the equals sign would be 180C1, which is just 180.

[DarkMagus]

Another way to solve it is use the binomial distribution.

4 successes, 100 trials, 1/221 probability of success.

In excel you'd type

=BINOMDIST(4,100,1/221,FALSE)

This gives a probability of 0.001064, or 1 in 940. This is how many times you get exactly 4 AA's in 100 hands.

You can also find the probability of getting 4 or more AA's by using the cumulative binomial distribution. Put in 3 as your number of successes and subtract this number from 1.

In excel:

=1-BINOMDIST(3,100,1/221,TRUE)

This gives 0.001164, or 1 in 859. I believe the 1 in 859 is the number you're looking for.

[Idiotex]

I knew all that crap in year 12 maths had some use... If only I'd played poker back then.