MHB Probability of Winning with 21 Cards: OK You Probabilitizers!

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The discussion focuses on the probability of winning with a specific set of 21 cards labeled from 1 to 11, where a winning hand consists of at least three different cards valued less than 6. Participants are challenged to calculate the probability of drawing five cards at random without replacement that meet this winning criterion. An example illustrates that combinations with repeated values do not count towards the winning condition. The thread concludes with a note that challenges with known solutions should be posted in a designated forum. The conversation emphasizes the importance of understanding the rules for determining winning combinations in this card game scenario.
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OK you probabilitizers:
21 cards are labelled: 1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,7,8,9,10,11
You pick 5 cards at random, no replacement.
To "win", you need at least 3 DIFFERENT cards < 6.
Example: two 3's counts as one 3; three 4's counts as one 4.
So, as example, 5,2,5,9,5 is not a winning combo...got that?
But 3,1,3,3,5 is a winner...ok?
What's the probability of picking a winning combo?​
 
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No takers? Solution:
Relabel the cards: 1a 1b 1c 2a 2b 2c 3a 3b 3c 4a 4b 4c 5a 5b 5c 6 7 8 9 10 11
so there are C(21, 5) equally likely hands.

Sort the hand. Then a winning combination must be one of the following.

A)

Two high cards and three different low cards
C(6,2) * C(5,3) * 3 * 3 * 3 = 150 * 27 = 4050
(The last factors of 3 are the selection of a, b, or c.)

B)

One high card, three different low cards, and a matching low card
C(6,1) * C(5,3) * 3 * 3 * 3 * 3 * 2 / 2 = 60 * 81 = 4860
(The division by 2 is to allow for the order of the match.)
One high card and four different low cards
C(6,1) * C(5,4) * 3 * 3 * 3 * 3 = 2430

C)

Two pairs of matching low cards, and a different low card
C(5,2) * 3 * 3 * C(3,1) * 3 = 810
(The middle factors of 3 are the omission of a, b, or c.)
A triple low card, and two different low cards
C(5,1) * C(4,2) * 3 * 3 = 270
Four different low cards, and a matching low card
C(5,4) * 3 * 3 * 3 * 3 * 4 * 2 / 2 = 1620
Five different low cards
3 * 3 * 3 * 3 * 3 = 243

So the probability of a winning combo is
(4050 + 4860 + 2430 + 810 + 270 + 1620 + 243) / C(21,5)
= 14283 / 20349
= 1587 / 2261
= 70.2 %
 
If you want to post challenge problems to which you already know the solution then we have a special forum for that. I'll move this there now.
 
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