UNIT 4 HOMEWORK SOLUTIONS
Math 102 & Core 143
1. Katie has a lot of hats; 5 black hats, 3 white hats, and 2 blue hats. Katie likes to suprise herself
by picking her hats at random (so that each hat is equally likely to be picked).
a) Katie doesn’t feel like wearing a black hat today, so given that she picked a black hat first, what
is the chance that she picks a white hat in the next two picks?
We use the not rule (we don’t want both of the next two picks each to be either black or blue).
1−
6
9
·
5
8= 1 −
10
24 =7
12
b) What is the chance that in the first three picks, at least one of the hats is not black?
We use the not rule (we don’t want all three picks to be black). 1−
5
10
·
4
9
·
3
8= 1 −
1
12 =11
12 .
2. There are ten socks a his drawer: one pair of black socks, two identical pairs of white socks and
four odd socks colored red, blue, tan and yellow. Fred picks his socks out of the drawer in a way so
that every sock is equally likely to be picked. Each of the following questions is a separate scenario.
a) What is the chance that the first two socks pulled out are the black pair? 2
10
·
1
9=1
45 .
b) Suppose Fred pulls out the first sock, but drops it under the bed before noticing the color. He
quickly pulls out another sock. What is the chance that the second sock is an odd sock ?
There is no new information here. Think of the chance that the second card in a deck is an ace. It
is 4
52 not 4
51 . If we threw the first card under the dresser, it would still be 4
52 . Hence, the answer
here is 4
10 .
c) If the first sock pulled out is white, what is the chance that he gets a white pair at or before the
fourth sock?
We use the not rule. We don’t want the second, third, and fourth socks all to be “not white.” Hence,
we have 1−
6
9
·
5
8
·
4
7= 1 −
10
42 =19
24 .
d) What is the chance that the first two socks are a pair?
The pair could be either black or white. Hence, we have 2
10
·
1
9+4
10
·
3
9=7
45 .
3. Jane programs her computer to randomly select music to wake her up with each morning.
Understanding truly great music, she selects a 20% chance for music by The Sugarplastic. What is
the chance that The Sugarplastic would be selected eight times in eight days?
This is the multiplication principle. Hence, the answer is !1
5"8.
4. Roll a pair of fair dice.
a) On one roll, what is the chance that we get at least one !
•?We use the not rule: 1−
5
6
·
5
6=11
36 .
b) On ten rolls, what is the chance that we get at least one roll that has at least one !
•?
This is the Not Rule again. Using our answer to (a), we see that the probability of not getting at
least one !
•is 1 −11/36 = 25/36. Hence, the answer is
1−#25
36$10
