Download There is no guarantee that the answers are correct.

There is no guarantee that the answers are correct.
Math 110 Review for Exam-2 Summer 2009
1. Three coins are tossed. Using ordered triples, give: (a) the sample space,
{hhh, hht, hth, htt, thh, tht, tth, ttt}
(b) the event E that at least two are heads,
{hhh, hht, hth, thh}
(c) the probability that at least two are heads.
1/2
2. A family has 3 children. Using g to stand for girl and b for boy, and using ordered triples
such as (b, g, g), find (a) the sample space (list all elements),
{bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}
(b) the event E that the family has exactly two daughters,
{bgg, gbg, ggb}
(c) p(E) = 3/8 and o(E) = 3 : 5.
(b) Find the probability that all children are of the same sex. Answer 1/4
(c) Is it more likely that all children are of the same sex or that both sexes are represented?
p(both sexes)= 3/4 > 1/4 = p(same sex).
3 Find the expected value for each $1 bet in roulette.
(a) single number bet, (b) red-number bet. (c) six-number bet.
The answer is always -2/38 = -.053.
3
8
5
4. If p(E) = , p(F 0 ) = and p(E ∪ F ) = ,
9
9
9
(a) find p(F ) = 69 = 32 , (b) find p(E ∩ F ) = 59 + 69 − 98 = 39 = 13 .
5. Find the expected value of a $1 bet in the following game:
A pair of dice is rolled. If the sum is greater than 10, the player wins $11. Otherwise the
player loses the bet. Would you play this game? Why or why not?
3
The event “sum is greater than 10” has 3 element. Expected value = 11 · 36
+ (−1) · 33
36 = 0.
This is a fair game. Over a long period of time you will neither lose nor win much.
6. Find the expected value of a $1 bet in the following game: You are being dealt one card
from a deck of 54 cards (standard 52-card deck plus two jokers). If your card is a diamond
you win $3. Otherwise you lose the bet. Would you play this game? Why or why not?
13
2
The event “card is diamond” has 13 element. Expected value = 3 · 54
+ (−1) · 41
54 = − 54 < 0.
Don’t play because of negative expected value.
7. Flashlights are being sold at the local salvage store at a very low price. Of the remaining
16 flashlights 7 have some defects. The defects are not visible to the eye. You decide to buy
three of the flashlights.
Experiment is picking three from a pool of 16. Size of sample space is 16 C3 = 560
Find the probability of the following: (a) All three flashlights are broken
=
7 C3
16 C3
=
35
1
=
560
16
(b) All three flashlights are in working order.
=
9 C3
16 C3
=
84
3
=
560
20
(c) At least one flashlight is in working order.
1−
1
15
= .
16
16
8. (a) Find the probability of winning first prize (six winning numbers) from a 6/46 lottery.
=
1
1
=
9, 366, 819
46 C6
(b) Find the probability of winning third prize (having exactly four of the six winning numbers) from a 6/46 lottery.
11, 700
6 C4 ·40 C2
=
=
9, 366, 819
46 C6
(c) Find the probability of having not a single winning number from a 6/46 lottery.
=
40 C6
46 C6
=
3, 838, 380
= 41%
9, 366, 819
(d) Find the probability of having at least one winning number from a 6/46 lottery.
=1−
3, 838, 380
= 59%
9, 366, 819
9.
An insurance company estimates that 13% of insured drivers of a certain type of
automobile report an accident in a given year. The average damage to the driver’s car per
accident is $2350. Use the expected value to determine how much the insurance company
should collect as premium for collision coverage?
Premium at least $305.50.
10.
The Department of Mathematics at the University of Southwestern North Dakota
consists of 19 full-time faculty members. 14 are men and 5 are women. A four-person hiring
committee must be selected by the chair. How many committees are possible if it must consist of the following?
(a) Two women and two men. 14 C2 ·5 C2 = 910. (b) Any mixture of men and women.
19 C4 = 3, 876. (c) At least one woman. 3, 876 −14 C4 = 2, 875. (d) At least one woman and
at least one men. 3, 876 −14 C4 −5 C4 = 2, 870.
11.
From a standard deck with 52 cards your are being dealt a hand consisting of 5
cards.
(a) How many different hands with exactly 4 spades are possible? 13 C4 ·39 C1 = 27, 885 (b)
How many different hands with 5 hearts are possible? 13 C5 = 1, 287 (c) How many different
hands in one suit are possible? 4 ·13 C5 = 5, 148 (d) How many hands with 2 hearts and
3 spades are possible? 13 C2 ·13 C3 = 22, 308(e) What is the probability of getting a hand
575,757
39 C5
without any hearts? 52
C5 = 2,598,960 = 22% (f) What is the probability of getting a hand
with at least one heart? 78% (g) What is the probability of getting a hand without any
1,712304
48 C5
kings? 52
C5 = 2,598,960 = 66%
12.
A pair of dice is rolled. (a) Using ordered pairs, list the elements of the event E
that the sum is 6. {(5, 1), (1, 5), (2, 4), (4, 2), (3, 3)} (b) Find probability that the sum is 6.
5/36