tryingtoexcelatmath
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- Mar 5, 2018
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In Bridge we have 4 players, and each get 13 cards.
The most symmetrical hand is the (4, 3, 3, 3) hand which is 4 of 1 card type (spades) 3 hearts 3 diamonds and 3 clubs. (the card type order does not matter)
So the video says:
" We will now count these hands, and to do that we will first specify which suit in your hand has the 4 cards. Once we choose that suit, there are 13 choose 4 choices of the 4 cards in that suit because there are 13 cards of each suit in the deck and we take 4 of them. It doesnt matter which order the cards are dealt. So it is a collection not a sequence. In the remaining suits there are 13 choose 3 choices for the cards in each suit so by the multiplication principle. the total number of different hands with a 4, 3, 3, 3, distribution is 4 * 13 choose 4 * 13 choose 3 * 13 choose 3 * 13 choose 3"
TLDR:
4 * 13 choose 4 * 13 choose 3 * 13 choose 3 * 13 choose 3
we have 67B different hands w/ a total bridge hands being 52 choose 13 = 635B, so answer is 67/635 = 10.5%
10.5% of the time we will get a 4, 3, 3, 3 hand
In the next example (4, 4, 3, 2): the video states:
"Let's work out the number of hands with a 4, 4, 3, 2 distribution, To do this, we first choose which of the 2 suits will have 4 cards"
My first question here is why are we choosing the first 2 suits with 4 cards? In the example above for 4,3,3,3 we just multiplied by 4 in the beginning and that was it.
"There are 4 suits, we choose 2 of them, so there are 4 choose 2 = 6 possible choices of those suits, and in each of those suits, we have to choose 4 of the 13 cards, so there are 13 choose 4 choices for those cards. Now we have 2 suits remaining, one of them has 3 cards the other has 2 cards. First we choose the one that has 3 cards and there are 2 possible choices for that. Then once we've chosen that there are 13 choose 3 choices for the cards we hold in that suit. And Finally there are 13 choose 2 choices for the card in the remaining suit. So by the multiplication rule.."
TLDR:
(4 choose 2) * (13 choose 4) * (13 choose 4) * (2 choose 1) * (13 choose 3) * (13 choose 2)
How come in the second example of 4,4,3,2 we need a (4 choose 2) and then another (2 choose 1) for the choice of suits.
I understand all of the (13 choose x) because we have 13 cards in that suit and we are choosing X cards from each suit.
But in the first example given by the video they only have a 4 in the beginning (which I am assuming is 4 choose 1, we have 4 suits to choose from and we choose 1 of them)
How come in the first 4,3,3,3 example we don't have a (3 choose 1 suit) for the remaining 3 suits?
But in the 4,4,3,2 example they had a (4 choose 2) and then a (2 choose 1) to choose the different suits.
Why is that? What is the difference between the 4, 3, 3, 3, example and the 4, 4, 3, 2 example?
Video URL:
The most symmetrical hand is the (4, 3, 3, 3) hand which is 4 of 1 card type (spades) 3 hearts 3 diamonds and 3 clubs. (the card type order does not matter)
So the video says:
" We will now count these hands, and to do that we will first specify which suit in your hand has the 4 cards. Once we choose that suit, there are 13 choose 4 choices of the 4 cards in that suit because there are 13 cards of each suit in the deck and we take 4 of them. It doesnt matter which order the cards are dealt. So it is a collection not a sequence. In the remaining suits there are 13 choose 3 choices for the cards in each suit so by the multiplication principle. the total number of different hands with a 4, 3, 3, 3, distribution is 4 * 13 choose 4 * 13 choose 3 * 13 choose 3 * 13 choose 3"
TLDR:
4 * 13 choose 4 * 13 choose 3 * 13 choose 3 * 13 choose 3
we have 67B different hands w/ a total bridge hands being 52 choose 13 = 635B, so answer is 67/635 = 10.5%
10.5% of the time we will get a 4, 3, 3, 3 hand
In the next example (4, 4, 3, 2): the video states:
"Let's work out the number of hands with a 4, 4, 3, 2 distribution, To do this, we first choose which of the 2 suits will have 4 cards"
My first question here is why are we choosing the first 2 suits with 4 cards? In the example above for 4,3,3,3 we just multiplied by 4 in the beginning and that was it.
"There are 4 suits, we choose 2 of them, so there are 4 choose 2 = 6 possible choices of those suits, and in each of those suits, we have to choose 4 of the 13 cards, so there are 13 choose 4 choices for those cards. Now we have 2 suits remaining, one of them has 3 cards the other has 2 cards. First we choose the one that has 3 cards and there are 2 possible choices for that. Then once we've chosen that there are 13 choose 3 choices for the cards we hold in that suit. And Finally there are 13 choose 2 choices for the card in the remaining suit. So by the multiplication rule.."
TLDR:
(4 choose 2) * (13 choose 4) * (13 choose 4) * (2 choose 1) * (13 choose 3) * (13 choose 2)
How come in the second example of 4,4,3,2 we need a (4 choose 2) and then another (2 choose 1) for the choice of suits.
I understand all of the (13 choose x) because we have 13 cards in that suit and we are choosing X cards from each suit.
But in the first example given by the video they only have a 4 in the beginning (which I am assuming is 4 choose 1, we have 4 suits to choose from and we choose 1 of them)
How come in the first 4,3,3,3 example we don't have a (3 choose 1 suit) for the remaining 3 suits?
But in the 4,4,3,2 example they had a (4 choose 2) and then a (2 choose 1) to choose the different suits.
Why is that? What is the difference between the 4, 3, 3, 3, example and the 4, 4, 3, 2 example?
Video URL:
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