Three players are each dealt, in a random manner, five cards from a deck containing 52 cards. Four of the 52 cards are aces. What is the probability that at least one person receives exactly two aces in their five cards?
Let $A_i$ represent the player $i$ with two aces where $i = 1,2,3$. The probability a player receives two aces is the following.
$$P(A_i) = \frac{{4 \choose 2}{48 \choose 3}}{{52 \choose 5}} \approx .0399$$
Then the probability at least one person receives exactly two aces is the following.
$$3 \cdot .0399 – 3 \cdot .0399^2 \approx .1149$$
Is this correct?
Best Answer
You have used Inclusion/Exclusion. That is a good idea, but the exclusion part needs to be recomputed. Let us, for example, calculate the probability that A and B each get exactly $2$ Aces. This is the probability A gets $2$ Aces, times the probability B gets $2$ Aces given that A got $2$. So B is getting her $5$ cards from a deck that has $2$ Aces and $45$ non-Aces. The probability she gets $2$ Aces given that A got $2$ is $\binom{45}{3}/\binom{47}{5}$. Quite small.