Chapter 5 Section review day 1 2016s.notebook
April 21, 2016
Weird dice Nonstandard dice can produce interesting distributions of outcomes.
Suppose you have two balanced, six-sided dice. Die A has faces with 2, 2, 2, 2,
6, and 6 spots. Die B has faces with 1, 1, 1, 5, 5, and 5 spots. Imagine that you
roll both dice at the same time.
(a) Find a probability model for the difference (Die A − Die B) in the total number
of spots on the up-faces.
(b) Which die is more likely to roll a higher number? Justify your answer.
6
18
24
Die A P(A>B) = ------- + -------- = ------36
36
36
Nov 22-7:36 PM
Race and ethnicity The Census Bureau allows each person to choose from a long list of
races. That is, in the eyes of the Census Bureau, you belong to whatever race you say you
belong to. Hispanic (also called Latino) is a separate category. Hispanics may be of any
race. If we choose a resident of the United States at random, the Census Bureau gives
these probabilities:25
0.149
0.851
0.045
0.130
0.813
0.012
1.000
> (a) Verify that this is a legitimate assignment of probabilities.
See above all total total is 1
> (b) What is the probability that a randomly chosen American is Hispanic?
P(H) = (0.001+0.006+0.139+0.003)=0.149
> (c) Non-Hispanic whites are the historical majority in the United States. What is the
probability that a randomly chosen American is not a member of this group?
Wc= person chosen is NOT (white non-hispanic)
P(Wc) = 1- 0.674 = 0.326
> (d) Explain why P(white or Hispanic) ≠ P(white) +P(Hispanic). Then find P(white or
Hispanic).
The events white and Hispanic are NOT disjoint so the general
addition formula must be used.
P(white
Hispanic) = 0.813 + 0.149 - 0.139 = 0.823
Nov 22-7:47 PM
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Chapter 5 Section review day 1 2016s.notebook
April 21, 2016
In 2012, fans at Arizona Diamondbacks home games would win 3 free tacos from
Taco Bell if the Diamondbacks scored 6 or more runs. In the 2012 season, the
Diamondbacks won 41 of their 81 home games and gave away free tacos in 30
of their 81 home games. In 26 of the games, the Diamondbacks won and gave
away free tacos. Choose a Diamondbacks home game at random.
(a) Make a Venn diagram to model this chance process.
Won
Taco
4
15
26
36
(b) What is the probability that the Diamondbacks lost and did not give away free
tacos?
36
P(lost ∩ TC) = ----- = 0.4444
81
(c) What is the probability that the Diamondbacks won the game or fans got free
tacos?
41
P(won
30
26
45
81
81
81
taco) = ----+ ----- - ------ = ---- = 0.5556
81
Nov 22-7:52 PM
Steroids A company has developed a drug test to detect steroid use by athletes. The test is
accurate 95% of the time when an athlete has taken steroids. It is 97% accurate when an
athlete hasn’t taken steroids. Suppose that the drug test will be used in a population of athletes
in which 10% have actually taken steroids. Let’s choose an athlete at random and administer
the drug test.
(a) Make a tree diagram showing the sample space of this chance process.
0.10
steroids
0.95
+
_
0.05
P(steroids ∩ +)= (0.10)(0.95)
0.095
P(steroids ∩ -)= (0.10)(0.05)
0.0051
all athletes
steroidsC
0.03
0.90
+
_
0.97
P(steroidsC ∩ +)= (0.90)(0.03)
0.027
P(steroidsC ∩ -)= (0.90)(0.97)
0.873
(b) What’s the probability that the randomly selected athlete tests positive? Show your work.
P(positive test result) = 0.095 + 0.027 = 0.122
(c) Suppose that the chosen athlete tests positive. What’s the probability that he or she actually
used steroids? Show your work.
0.095
P(steroid use I positive test result) = ---------- = 0.7787
0.122
Nov 22-7:53 PM
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Chapter 5 Section review day 1 2016s.notebook
April 21, 2016
R5.7. Mike’s pizza You work at Mike’s pizza shop. You have the following
information about the 7 pizzas in the oven: 3 of the 7 have thick crust and 2 of
the 3 thick crust pizzas have mushrooms. Of the remaining 4 pizzas, 2 have
mushrooms. Choose a pizza at random from the oven.
(a) Make a two-way table to model this chance process.
Thick ThickC
Mush
2
2
4
Mush C
1
2
3
3
4
7
(b) Are the events “getting a thick-crust pizza” and “getting a pizza with
mushrooms” independent? Explain.
There are three ways to check ... you only need to check one way
P(T
M) = P(T) P(M) ? P(T
P(T l M) = P(T) ?
P(T l M ) = 0.5 P(T) = 0.4286
P(M l T) = P(M) ?
P(M l T) = 0.667 P(M) = 0.5714
The event thick crust and pizza with mushrooms are NOT INDEPENDENT
because none of the above probabilities are equal.
(c) You add an eighth pizza to the oven. This pizza has thick crust with only
cheese. Now are the events “getting a thick-crust pizza” and “getting a pizza with
mushrooms” independent? Explain.
Thick ThickC
Mush
2
2
4
Mush C
2
2
4
4
4
8
There are three ways to check ... you only need to check one way
P(T
M) = P(T) P(M) ? P(T
P(T l M) = P(T) ?
P(T l M ) = 0.5 P(T) = 0.5
P(M l T) = P(M) ?
P(M l T) = 0.5 P(M) = 0.5
The event thick crust and pizza with mushrooms are INDEPENDENT
because all of the above probabilities are equal.
Nov 22-7:54 PM
R5.8. Deer and pine seedlings As suburban gardeners know, deer will eat
almost anything green. In a study of pine seedlings at an environmental
center in Ohio, researchers noted how deer damage varied with how much
of the seedling was covered by thorny undergrowth:26
211
234
221
205
209
662
871
(a) What is the probability that a randomly selected seedling was damaged
by deer?
209
P(seedling damage by deer) = ------- = 0.24
871
(b) What are the conditional probabilities that a randomly selected seedling
was damaged, given each level of cover?
60
P(damage I none) = -------- = 0.2843
211
76
P(damage I <1/3) = -------- = 0.3248
234
44
P(damage I 1/3 to 2/3) = -------- = 0.199
221
29
P(damage I >2/3) = -------- = 0.1415
205
(c) Does knowing about the amount of thorny cover on a seedling change
the probability of deer damage? Justify your answer.
Yes, it appears that in general (except < 1/3) the more thorny cover
the less damage to the pine seedlings. The thorns deter the deer!
Nov 22-8:09 PM
3
Chapter 5 Section review day 1 2016s.notebook
April 21, 2016
R5.9. A random walk on Wall Street? The “random walk” theory of stock
prices holds that price movements in disjoint time periods are independent
of each other. Suppose that we record only whether the price is up or down
each year, and that the probability that our portfolio rises in price in any one
year is 0.65. (This probability is approximately correct for a portfolio
containing equal dollar amounts of all common stocks listed on the New
York Stock Exchange.)
(a) What is the probability that our portfolio goes up for three consecutive years?
P(up for three consecutive years) = (0.65)3 = 0.2746
(b) What is the probability that the portfolio’s value moves in the same
direction (either up or down) for three consecutive years?
P(same direction for three years) = (0.65)3 + (0.35)3 = 0.3175
Nov 22-8:11 PM
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