Probability

Question

In the card game bridge, the 52 cards are dealt out equally to 4 players—called East,
West, North, and South. If North and South have a total of 8 spades among them,
what is the probability that East has 3 of the remaining 5 spades?

Comments

$\Large \frac{\binom{5}{3} * \binom{21}{10}}{\binom{26}{13}}$ = 0.3391

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$\frac{\binom{5}{3}\times \binom{39}{10}}{\binom{46}{13}}$

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srestha sample space is reduced as it is mentioned that "52 cards are dealt out equally to 4 players", hence the cards are already distributed and North and South have a total of 8 spades among them. That means in remaining 26 cards which are with East and West we need to calculate probability. No matter what non spades are with North and South they are not gonna interfere with the combinations of East and West.

Answer 1

North and South already have 26 cards out of which 8 are spades, so remaining 26 cards among East and West have 5 spade remaining. So what 26 cards North and South have it doesn't matter, let cut them off and reduce our working sample space.

NOW, we have only 26 cards in which 5 are spades and two players E and W.

so to give East 3 cards out of 5 there are C(5,3) ways and remaining 10 will be given from other than those 5 cards(spades) to EAST in C(26,10) ways.

And to choose 13 cards among 26 cards for EAST we have C(26,13).

 
$\LARGE P=\frac{\binom{5}{3}\binom{21}{10}}{\binom{26}{13}}$

Comments

Thanks....Got it