Question

1. Suppose you have 2 nickels, 3 dimes, and 1 quarter in a coin purse. You choose two coins from the purse at random. What is the probability you draw out at least 20 cents?

2. A three of a kind in a five-card poker hand is obtained by having exactly three of the same type of card (3 Js or 3 4s). What is the probability that a randomly selected five-card hand is a three of a kind?

3. A committee of 3 is to be selected from 4 Democrats and 7 Republicans. What is the probability that the committee has exactly 2 Democrats given that it has at most one Republican?

Answer 1

2)

Total cards = 52          
number of possible outcomes of selecting 5 cards = 52C5 = 2598960          

Three of a kind-   3 cards of one denomination and fourth card of a second denomination and fith card of a third denomination                              
there is 13 possible cases of one denomination                                  
from each set(4 cards), 3 cards are drawn in 4C3 ways                                  
for second and third denomination, 12C2 ways are there                                  
now, picking 2 cards from 12 cards , and have 4 identical each                                  
so, total possible outcome= 13*4C3 * 12C2 * 4² =       54912                          
                                  
P(three of a kind) = 54912/2598960 =    0.02113                              

3)

P(at most one republican) = P(0 republican, 3 democrat) + P(1 republican, 2 democrat) = 7C0*4C3/11C3 + 7C1*4C2/11C3 = 0.2788

probability that the committee has exactly 2 Democrats given that it has at most one Republican =(7C1*4C2/11C )/0.2788 = 0.9130