1. In poker, we choose a five-card hand randomly, and without replacement, from
ID: 3176182 • Letter: 1
Question
1. In poker, we choose a five-card hand randomly, and without replacement, from a standard 52-card deck. Find the probability that our hand
(a) contains exactly 3 cards in the same suit
(b) contains at least 3 hearts, but no spades.
2. Our Xena fan club consists of 32 science majors, 27 engineering majors and 15 arts majors. We must select a committee of 10 members. If we choose randomly, find the probability of getting
(a) exactly 5 science majors, 3 engineering majors and 2 arts majors
(b) at least 2 engineering majors (try to do this without adding 9 terms together).
3. In a survey, 100 participants were asked if they like cake, pie or ice cream. They were free to pick more than one, or none. In fact, 31 liked cake, 19 liked pie, 27 liked ice cream, 8 liked cake and pie, 14 liked cake and ice cream, 9 liked pie and ice cream and 6 liked all three. (As usual, when we say that 8 liked cake and pie, that would include the 6 who liked all 3.)
(a) Find the probability that a randomly selected participant liked pie but not ice cream.
(b) Given that a randomly selected participant liked pie, find the probability that he did not like cake.
Explanation / Answer
1) 52 cards ,
4 suits,each suit 13 cards.
(a) contains exactly 3 cards in the same suit
We choose one suit from which 3 cards will be there in 4C1 ways. Then we choose 3 cards in 13C3 ways.
Now for rest two card ,we have 39 cards to choose from, hence 39C2 ways .
hence total number of ways = 4C1 * 13C3 * 39C2 = 847704
total number of ways without any condition = 52C5 = 2598960
required probability = 847704/2598960 = 0.3261
b)(b) contains at least 3 hearts, but no spades.
case , 1) 3 heart , 2)4 heart 3) 5 heart
number of ways of
1) 13C3 * 26C2 = 286 * 325 = 92950 { we have to choose 2 cards,but we can't choose from spade also ,hence 52-13-13 = 26}
2)13C4*26C1 = 715*26 = 18590
3)13C5 = 1287
total number of ways with given condition = 92950+18590+1287 = 112827
hence required probability= 112827/ 2598960 = 0.0434123649
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