Mathematics 1550H Assignment #2 | Complete Solution

Mathematics 1550H { Introduction to probability Trent University, Winter 2016

Assignment #2
Hands across the deck
Recall that a standard 52-card deck has four suits, namely ~, }, |, , each of which has
thirteen cards, namely A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2. We will be dealing with ve-card hands
drawn at random, all at once, from the deck.
1. What is the probability that any particular ve-card hand will be drawn? [1]
Suppose that after a ve-card hand is drawn, the cards in it are put back in the deck and
another ve-card hand is drawn.
2. What is the probability that the two hands have no card in common? [1]
3. What is the probability that the two hands have exactly one card in common? [1]
4. What is the probability that the two hands have at least one card in common? [1]
5. What is the probability that the two hands have at least three cards in common? [1]
A straight is a hand in which the cards are in consecutive order by rank. For the purposes
of the following question, going around the end of the rank order is not allowed. (For example,
4 3 2AK would not count as a straight.)
6. Suppose you draw a ve-card hand randomly from the deck and get four cards that that would
make a straight if you could replace the fth card. (e.g. J 10 9 8 3 or K 7 6 4 3). If you are
allowed to discard the fth card and draw one at random from the remaining 47 cards in the
deck, what is the probability that your modi ed hand will be a straight? [3]
Hint: There are several cases to consider : : :
A
ush is a hand in which all the cards are from the same suit.
7. Suppose you draw a ve-card hand randomly from the deck. What is the probability that this
hand is a
ush? [1]
8. Suppose you draw a ve-card hand randomly from the deck. What is the probability that this
hand is both a straight and a
ush? [1]