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February 1st, 2018, 11:30 PM   #1
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What is the probability of being dealt a bridge hand with exactly 4 honour cards and exactly 4 cards from the 5 through 10?
Each hand is dealt 13 cards.
Honour cards: 20 cards (A, K, Q, J, 10)
From 5 through 10: 24 cards (10, 9, 8, 7, 6, 5)
Others: 12 cards (4, 3, 2)
I am thinking about 5 cases.
Case1: No 10: (16C6*20C4*12C5)/52C13.
Case2: One 10: (4C1*16C3*20C3*12C6)/52C13.
Case3: Two 10's: (4C2*16C2*20C2*12C6)/52C13.
Case4: Three 10's:
Case5: Four 10's:
Then add them up.
But I have a feeling that I am doing it wrong.
Please help. Thank you.

Last edited by skipjack; February 2nd, 2018 at 04:48 AM.
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February 2nd, 2018, 12:23 AM   #2
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this is a multinomial

3 classes:
Honor cards: 16 (I'm assuming 10 isn't an honor card as you labeled it)
5 - 10: 24
Others: 12

$\dfrac{\dbinom{16}{4}\dbinom{24}{4}\dbinom{12}{5} }{\dbinom{52}{13}} \approx 0.024$

Last edited by romsek; February 2nd, 2018 at 12:26 AM.
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February 2nd, 2018, 06:33 AM   #3
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Thanks for your help but 10 is one of the honour cards that's why this question has multiple cases. I know it has multiple cases but but I don't know how to solve.
10 is overlapped between honour group cards and from the 5 to 10 group cards.
Please help.
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February 2nd, 2018, 09:24 AM   #4
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ok let's use 4 classes then

honor cards: 16 J-A
10's: 4
5-9: 20
others:12

Now apply your cases (all examples are divided by 52C13)

0 10s : (16C4)(4C0)(20C4)(12C5)
1 10 : (16C3)(4C1)(20C3)(12C6)
2 10s : (16C2)(4C2)(20C2)(12C7)
3 10s : (16C1)(4C3)(20C1)(12C8 )
4 10s : (16C0)(4C4)(20C0)(12C9)

and add all these

you were quite close, ring back if you don't understand all the numbers
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February 2nd, 2018, 02:26 PM   #5
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I have just gotten exactly what you did. Thank you very much for your help. I really appreciate for your help.
I am quite new here. Where can I give you a thank.
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February 2nd, 2018, 02:27 PM   #6
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Quote:
Originally Posted by Shanonhaliwell View Post
I have just gotten exactly what you did. Thank you very much for your help. I really appreciate for your help.
I am quite new here. Where can I give you a thank.
a thanks link should show up on the right of a post as you mouse in that direction
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