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Figuring the Odds of 3 Cards

Date: 08/07/97 at 01:00:37
From: Larry Ponder
Subject: Figuring the odds of 3 suited cards

In the card game Super Seven, how do you figure the odds against you 
of receiving three sevens?

6 deck shoe = 24 sevens  288 non-sevens.
   24 / 288 = 12         12 to 1 for the first seven. Right?

Now the shoe has only 5 of the same suit sevens, 323 of non-samesuit.
How do you figure the odds of getting the second suited seven and then 
how do you figure the odds of getting the third suited seven?

Thank you,
Larry Ponder

Date: 08/07/97 at 18:42:23
From: Doctor Tom
Subject: Re: Figuring the odds of 3 suited cards

Hi Larry,

You're on the right track.  It's easier to work with probabilities,
however.  For example, instead of saying "12 to 1", it's easier
to think "1 chance in 13".  It's equivalent - it means out of 13
tries, you get 1 win and 12 losses, which is what "12 to 1" means.
The advantage of probabilities is that there are nice rules about
when they can be added or multiplied to give other probabilities.

So let's calculate the probability of winning. If you only had to
pick 1 seven to win, the probability of winning would be 1/13.

Then, as you pointed out, after you've picked the first 7, there
are only 5 cards that will help you out of the 287 remaining, so
given that you made it past the first draw, you will make it past
the second draw only 5 times out of every 287, or 5/287 of the time.

For the third, with similar reasoning, you'll win 4/286 of the time.

To win, you have to get past the first, second, and third draw
successfully, and to get the probability of that, simply multiply
the three probabilities:

probability of win = (1/13)(5/287)(4/286) = 10/533533 = .000018742.

Or another way to look at it is that out of every 533533 times you
play the game, you'll win 10 times and lose 533523 times.

-Doctor Tom,  The Math Forum
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Associated Topics:
High School Probability

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