Calling all Statsmeisters

[divemaster]

There is a probability scenario that I have been mulling over lately. My logical mind and my intuition are battling over this.

Let's say that the odds of being dealt a pat flush poker hand are 1:60 (I'm not sure of the exact odds, but this will do for illustration). A dealer agrees to deal you 60 hands.

Let's also say that there is a raffle where 60 individuals place their names into the proverbial hat, one to be drawn.

Which of these gives you the best chance of winning?

My intuition says the raffle because there is a guaranteed winner. There is no guarantee that 60 poker hands will produce a flush.

If my life were dependent on winning, I'd choose the raffle and rather participate in a contest that depends on the law of large numbers.

But my logical mind says "Hey, a 1 in 60 chance is a 1 in 60 chance. No difference."

Is there a difference between 1:60 and 1:60 as I've described it?

[pldennison]

Are you talking about the dealer reshuffling the deck every time? If so, I believe that, since each hand is an independent trial, with duplication allowed, you could go a lot longer than 60 hands without a flush. But I don't remember why. :) Dex?

[divemaster]

Sure, reshuffle each time. I'm assuming a "perfect" contest. No bias in the deck; no bias in the raffle draw.

One certainly could go well over the theoretical number of hands (60, in this case) before getting that flush. Of course, you could get it on the second draw.

My question is geared toward whether that uncertainty is worth putting your life on the line vs. a 1:60 chance at a sure winner.

[Omniscient]

Flaw in your camparisons.

I understand that your saying that given a raffle with 60 enties has 60:1 odds. Thats right, but it doesn't translate to say given 60 cracks at it your garaunteed to win. This implies that you leave out the first drawing stub and pick again. Now the odds have changed to be 59:1 and so on. With the card game every hand the odds are 60:1, but with every draw from the raffle the odds get better. Now if you said that you threw the loser back into the raffle after the loss, and then redrew the odds remain 60:1. You can now see that there is no garauntee in either case that you will win once after 60 attempts. Clear?

[divemaster]

Thanks Omni! :)

I know I was missing something but I just couldn't get a grasp on it. Hey, it's Friday afternoon and I'm still at work with a fried noggin. :(

[Lumpy]

If you have a 1 in 60 chance of winning any single round, and you go 60 rounds, your chances of winning are as follows:

Let's say that the chance of losing any particular draw is 59 out of 60. So in other words, what is the chance you would continue to lose all sixty rounds? Your chance would be (59/60)^60. With a substantial probability of rounding errors, my calculator gives me the answer of .3587.

So you would have only about a 36% chance of losing all 60 rounds, meaning a 63% chance of getting a winning hand at least once.

[Lumpy]

Oops, truncated the wrong way. I mean a 67% chance of winning at least once.

[Lumpy]

D-OH! 64%!