Download Basic Probability Statistics

NAME: ANSWER KEY
INSTRUCTOR: Dr. Bathi Kasturiarachi
Math 30011
Spring 2009
Basic Probability & Statistics
Exam 2 { Part I { Sections (Chapter 4, Chapter 5)
SOLUTIONS
1.
What is the set of a single possible outcome of a probability experiment called?
(a). statistic
(b). sample space
(c). event
(d). Venn diagrams
Answer is (c).
2.
It is estimated that Lebron James makes 160 out of 200 free throws. We would estimate the probability
that Lebron will make his next free throw to be:
(a). 0.16
(b). 50-50, either makes it or doesn't
(c). 0.80
(d). 1.2
If the question was worded as: \We would estimate the probability that Lebron will make his next free
throw to be:" the answer is (c).
If the question was worded as: \The chance that Lebron will make his next free throw is:" the answer is
(b).
3.
A coin is tossed 5 times. What is the probability of getting at least one head?
P (at least one head) = 1
= 1
4.
P (all tails)
1 5
1
31
=1
=
' 0:968 75
2
32
32
The outcome of event A does not in uence the outcome of event B. Events A and B are called:
(a). mutually exclusive
(b). dependent
(c). disjoint
(d). independent
Answer is (d).
5.
In the game of roulette, a wheel consists of 38 slots numbered 0; 00; 1; 2; 3; ; 35; 36. The odd numbered
slots are red, and the even numbered slots are black. The numbers 0 and 00 are green. To play the
game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots.
1
(a) What is the probability that the metal ball lands on green or red?
P (red [ green) = P (red) + P (green) P (red \ green)
18
2
0
20
10
=
+
=
=
' 0:526 32
38 38 38
38
19
(b) What is the probability that the metal ball does not land on green?
P (not green) = 1
= 1
6.
P (green)
36
18
2
=
=
' 0:947 37
38
38
19
A drawer contains six black socks, four brown socks, and two green socks. Suppose that two socks are
drawn from the drawer without replacing the rst. Carefully draw a tree diagram and determine the
probability of getting two brown socks.
You should be able to draw the tree diagram easily. Remember to record all the probabilities. Let A be
the event that the rst drawn is brown, and B be the event that the second drawn is brown. Then we need,
P (A \ B) = P (A) P (BjA) =
7.
4 3
1
=
' 0:0909
12 11
11
If you draw an M& M candy from a bag of M& M candies, the candy you will draw is one of six colors.
The probability of drawing each color depends on the proportion of each color among all candies made.
Assume the probabilities are as given below.
Color
Probability
Brown
0.3
Red
0.3
Yellow
?
Green
0.1
Orange
0.1
Tan
0.1
(a) The probability of drawing a yellow candy is:
P (yellow) = 0:1
(b) The probability that you will NOT draw a red candy is:
P (not red) = 1
0:3 = 0:7
(c) The probability that you will draw either a brown OR green candy is:
P (brown [ green) = P (brown) + P (green) = 0:3 + 0:1 = 0:4
8.
Compute the probability that at least 1 male out of 1000 aged 24 years will die during the course of the
year if the probability that a randomly selected 24-year-old male survives the year is 0.9985.
P (at least one male dies) = 1
9.
P (all 1000 males survive) = 1
0:99851000 ' 0:777 12
A company is testing a new medicine for migraine headaches. In the study, 150 women were given the
new medicine and 100 women were given a placebo. Each participant was directed to take the medicine
when the rst symptoms of a migraine occurred and then to record whether the headache went away
within 45 minutes or lingered. The results are recorded in the following table.
Given medicine
Given placebo
Headache went away
132
56
Headache lingered
18
44
(a) If a study participant is selected at random, what is the probability that her headache went away
94
within 45 minutes? 188
250 = 125 ' 0:752
2
(b) If a study participant is selected at random, what is the probability that she was given the placebo
56
28
and her headache went away within 45 minutes? 250
= 125
' 0:224
(c) If a study participant is selected at random, what is the probability that she was given the placebo
or her headache went away within 45 minutes?
P (placebo [ headache went away) = P (placebo) + P (h.w.a.) P (placebo \ h.w.a)
100 188
56
116
=
+
=
' 0:928
250 250 250
125
10.
A UPS delivery route must include stops in ve locations in a city. How many di erent routes are
possible? Explain.
Use the factorial rule to obtain the number of di erent routes possible as = 5! = 120.
11.
There are 9 members on the board of directors at Akron General Hospital. If they must select a
chairperson, vice-chairperson, and a secretary, how many possibilities are there? Show all the work.
Use the permutations rule to obtain the number of possibilities as = 9P 3 =
Questions # 12 & # 13 are connected
9!
6!
= 7 8 9 = 504.
About 13% of the population is left handed.
12.
If two people are randomly selected, what is the probability both are left handed?
P (both left handed) = 0:132 ' 0:016 9
13. If two people are randomly selected, what is the probability at least one is right handed?
P (at least one is right handed) = 1
14.
P (both left handed) = 1
0:132 ' 0:9831
In the Illinois Lottery, a large urn contains balls numbered 1 to 54. From this urn, 6 balls are randomly
chosen without replacement. For a $ 1 bet, a player chooses two sets of 6 numbers. To win, all six
numbers (in either set) must match those chosen from the urn. The order in which the balls are selected
does not matter. What is the probability of winning?
P (winning) = 2
: 7: 743 8
10
1
2
=
= 7: 743 8
54C6
25827165
10
8
= 0:000000077438
8
15. Does the following determine a probability distribution? Explain.
x
1
2
3
4
16.
P (x)
0.037
X
P (x) = 0:977 6= 1:
0.200 No. Since
0.444
0.296
According to the National Endowment of Arts, 21% os U.S. women attended a musical play in 2003.
In a random sample of 15 U.S. women, what is the probability that exactly 5 have attended a musical
play in 2003? [HINT: You may assume the distribution is binomial, and use formula.]
P (X = 5) = (15C5)(0:215 )(0:7910 ) ' 0:1161
3
17.
A binomial distribution has n = 9 and p = 13 . Find the mean and standard deviation of this distribution.
mean :
standard deviation :
18.
1
3
= np = 9
=
p
npq =
r
=3
9
12 p
= 2
33
1:41
Use the binomial probability Table B to nd the probability of getting at least three correct responses
among 5 di erent requests from AT& T directory assistance. Assume that in general, AT& T is correct
90% of the time. That is, nd P (X
3). Treat this problem as a binomial distribution with success
probability as p = 0:90, and n = 5; q = 0:10. Use Table B.
P (X
3) = P (X = 3) + P (X = 4) + P (X = 5) = 0:073 + 0:328 + 0:590 = 0:991
19. When randomly selecting a jail inmate convicted of DWI (driving while intoxicated), the probability
distribution for the number X of prior DWI sentences is as described in the accompanying table. Find
the mean and standard deviation of this distribution.
X
0
1
2
3
P (X)
0.512
0.301
0.132
0.055
X P (X)
0.000
0.301
0.264
0.165
0.73
mean =
0:8894.
20.
P
X2
0
1
4
9
X 2 P (X)
0
0.301
0.528
0.495
1.324
X P (X) = 0:73 and standard deviation
=
pP
X 2 P (X)
2
=
p
1:324
(0:73)2
An investor has 0.60 probability of making a $ 20000 pro t and a 0.40 probability of su ering a $ 25000
loss. What is the expected value? Should she make the investment based on the expected value?
Expected Value E(X) = (20000)(0:6) + ( 25000)(0:4) = $2000. Based on the expectation, she should
invest.
4