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1. From a standard deck of 52 cards, the four Aces, four2 's, and four 3's are drawn, forming a smaller deck of 12 cards. All 12 cards are dealt at random to four players, so that each player gets three cards. What is the probability that each player's hand consists of an Ace, a 2, and a 3?

 

2. In a certain lottery, three white balls are drawn (at random) from ten balls numbered from 1 to 10, and one red SuperBall is drawn (at random) from ten balls numbered from 11 to 20. When you buy a ticket, you select three numbers from 1 to 10 and one number from 11 to 20. To win a super prize, the numbers on your ticket must match at least two of the white balls or must match the red SuperBall.

If you buy a ticket, what is your probability of winning a super prize?

 Mar 1, 2020
 #1
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\(P[\text{player 1 gets A, 2, 3}]= \dfrac{\dbinom{4}{1}\dbinom{4}{1}\dbinom{4}{1}}{\dbinom{12}{3}} = \dfrac{16}{55}\\ P[\text{player 2 gets A, 2, 3|player 1 did}] = \dfrac{\dbinom{3}{1}\dbinom{3}{1}\dbinom{3}{1}}{\dbinom{9}{3}}=\dfrac{9}{28}\\ P[\text{player 3 gets A,2,3|1 and 2 did}] = \dfrac{\dbinom{2}{1}\dbinom{2}{1}\dbinom{2}{1}}{\dbinom{6}{3}}= \dfrac{2}{5}\\ p= \dfrac{16}{55}\dfrac{9}{28}\dfrac{2}{5} = \dfrac{72}{1925}\)

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 Mar 1, 2020
 #2
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\(p=\dfrac{\dbinom{3}{2}\dbinom{8}{1}}{\dbinom{10}{3}} + \dfrac{1}{10}-\dfrac{\dbinom{3}{2}\dbinom{8}{1}}{\dbinom{10}{3}} \cdot \dfrac{10}{10}= \dfrac{7}{25}\)

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 Mar 1, 2020

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