Question

In this card game, a player is dealt 10 cards. A player wins $1,000,000 if he is dealt 8 hearts. Use combinatorics to set up your problem a. Calculate the probability a player is dealt exactly 8 hearts b. Calculate the probability a player is dealt exactly 8 hearts if the first tweo cards he was dealt were diamonds. 2

Answer 1

Assume a standard deck of 52 cards with 4 suits of 13 cards each.

a. There are $\dbinom{13}8\dbinom{39}2$ ways of being dealt a hand consisting of 8 hearts and 2 non-hearts, so the probability of being dealt such a hand is

$\dfrac{\dbinom{13}8\dbinom{39}2}{\dbinom{52}{10}}\approx0.0000602823$

b. This time, the non-hearts specifically belong to the suit of diamonds, for which there are $\dbinom{13}2$ ways of getting drawn, so the probability is

$\dfrac{\dbinom{13}8\dbinom{13}2}{\dbinom{52}{10}}\approx0.0000063455$