Download MATH 138 F11 Quiz 6 Solutions 1 1. Suppose one picks a card at

MATH 138 F11 Quiz 6 Solutions
1
1. Suppose one picks a card at random from a deck of 52 cards (having four suits hearts,
diamonds, spades, clubs).
a) What is the sample space (as a set)?
Solution The sample space is the set S of 52 cards in a deck.
b) Describe the event (as a set) that the chosen card is a face card which is red.
Solution The event E here is the set of red face cards - J♦,Q♦, K♦, J♥,Q♥,
K♥.
c) What is the probability that one chooses a face card which is red?
Solution The probability is P (E) =
N (E)
N (S)
=
6
52
=
3
.
26
2. Two faces of a six-sided die are painted red, two are painted blue, and two are painted
yellow. The die is rolled three times, and the colours that appear face up on the first,
second, and third rolls are recorded.
a) How many outcomes are there where the colour red appears exactly once?
Solution There are three ways the colour red can appear once, either in the
first, second, or third roll. For each of these three possibilities, there are four
possibilities for the other two rolls (either BB, BY, YB, or YY). Thus by the
multiplication rule there are 3 × 4 = 12 rolls where red appears once.
b) Find the probability that the colour red appears exactly once.
Solution By the multiplication rule, there are 3 × 3 × 3 = 27 possible rolls.
= 94 .
Therefore, the probability of getting red exactly once is 12
27
3. One urn contains three black balls (labeled B1 , B2 , B3 ). A second urn contains one
black ball (labeled B4 ) and two white balls (labeled W1 and W2 ). We perform an
experiment where first one of the two urns is chosen at random. Next a ball is randomly
chosen from the urn (without replacement). Then a second ball is chosen at random
from the same urn.
a) Construct a possibility tree for the experiment.
MATH 138 F11 Quiz 6 Solutions
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b) What is the probability that two black balls are chosen?
Solution From the above tree, there are 6 possible ways one get choose two black
6
= 12 .
balls, out of 12 possibilities. Thus the probability is 12
Reasoning another way, if we choose the first urn, we are guaranteed to pick to
black balls, whereas if we choose the second urn, we are guaranteed not to choose
two black balls. Thus the probability of choosing two black balls is 21 .
Bonus Suppose you are appearing on a game show with a prize behind one of six closed
doors: A,B,C,D,E, and F. If you pick the right door, you win a prize. You pick door A.
The game show host then opens one of the other doors and reveals that there is no prize
behind it. Then the host gives you the option of staying with your original choice of door
A or switching to one of the other doors that is still closed. If you switch to another door,
what is the probability that you will win the prize?
Solution Suppose you pick door A and you are shown door B with nothing behind it.
Then you pick one of the doors C, D, E, F.
If the prize is behind A, you lose in each of the four cases (when you pick either C,D,F,
or E).
If the prize is behind C, you lose in three cases (picking D,E, or F) and win in one case
(picking C).
If the prize is behind D, you lose in three cases, and win in one case.
If the prize is behind E, you lose in three cases, and win in one case.
If the prize is behind F, you lose in three cases, and win in one case.
There are 5 × 4 = 20 cases in total, and you win in exactly 4 of them. Therefore the
4
= 51 .
probability of winning is 20