# Probabilities of cards distribution

Hw do i estimate which pattern is the most likely in the distribution of the number of cards in each suit in a hand among 5-3-3-2; 4-4-3-2 and 4-3-3-3. This is a Bridge Game related problem

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Try to count out the number of possible hand types.

5-3-3-2 can be chosen in $\displaystyle 12 \ \binom{13}{5} \ \binom{8}{3} \ \binom{5}{3} = 8648640$

(the 12 factor comes from 5=3=3=2, 3=5=3=2, etc).

4-4-3-2 : $\displaystyle 12 \ \binom{13}{4} \ \binom{9}{4} \ \binom{5}{3} = 10810800$

4-3-3-3 : $\displaystyle 4 \ \binom{13}{4} \ \binom{9}{3} \ \binom{6}{3} = 4804800$

So 4-4-3-2 is most likely.

Of course, there might be errors and I am counting a particular suit split, which I suppose, should give the same proportion of probabilities as the hand pattern you seek.

Hope that helps.

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