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Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
Hierarchy of management that covers different levels of management
Β
Practice Test 2 Solutions
1. 1
Statistics, Sample Test (Exam Review) Solution
Module 2: Chapters 4 & 5 Review
Chapter 4 Probability
Chapter 5: Discrete Probability Distribution
Chapter 4 Probability
1. Definitions:
a. A simple event is an outcome or event that cannot be further broken down.
b. A sample space is a procedure that consists of all possible sample events.
c. If two events are mutually exclusive, the probability that both will occur is
( ) 0
P A B
ο =
d. The probability of an event is always:
a. between 0 and 1 0 β€ P(A) β€1
e. The sum of probabilities of all final outcomes of an experiment is always
1
( ) 1
n
i
i
P x
=
=
ο₯
2. Answer the following:
a. The number of Combinations of n items selected n at a time is
nCr =
π!
(πβπ)!π!
nCn =
π!
(πβπ)!π!
=
π!
0!π!
=
π!
π!
= 1
b. The number of Permutations of n items selected 0 at a time is
nPr =
π!
(πβπ)!
nPo =
π!
(πβπ)!
=
π!
π!
= 1
c. A pizza parlor offers 10 different toppings; how many four topping pizzas (different
toppings) are possible?
210 four topping pizzas are possible
Order is not important β combination is used
nCr =
π!
(πβπ)!π!
10C4 =
10!
(10β4)!4!
=
10π₯9π₯8π₯7
4π₯3π₯2
= 210
2. 2
d. How many 6-letter code words can be made from the 26 letters of the alphabet if no
letter can be used more than once in the code word?
165, 765, 600 6-letter code words can be made
Order is important, no letter can be used more than once β permutation is used
26P6 =
26!
(26β6)!
=
26π₯25π₯24π₯23π₯22π₯21
1
= 165,765, 600
3. Answer the following:
a. A quiz consists of 3 true-false questions, how many possible answer keys are there?
Write out the sample space and tree diagram.
8 possible answer keys
23
= 8
Sample Space: {TTT, TFF, FTF, FFT, FTT, TFT, TTF, FFF}
b. The sample space for tossing 5 coins consists of how many outcomes? Write out the
sample space.
Each coin has 2 outcomes: 25
= 32
Since there are only 2 possible outcome for each coin, Tail (T) or Head
(H), and there are 5 coins, Number of outcomes will be = 2 x 2 x 2 x 2 x 2
= 25 = 32
In general: Size of Sample Space = (# of outcomes per stage) # of stages
Sample Space:
{HHHHH, TTTTT, HHHHT, HHHTH, HHTHH, HTHHH, THHHH,
TTTTH, TTTHT, TTHTT, THTTT, HTTTT, HHHTT, HHTTH, HTTHH,
TTHHH, HHTHT, HTHHT, THHHT, HTHTH, THTHH, THHTH, TTTHH,
TTHHT, THHTT, HHTTT, TTHTH, THTTH, HTTTH, THTHT, HTHTT,
HTTHT}
4. A random sample of 100 people was asked if they were for or against the tax increase on
rich people. Of 60 males 45 were in favor, of all females 22 were in favor. Write the
5. 5
P(all 3 red) =
14
50
Γ
13
49
Γ
12
48
=
2184
117600
=
13
700
= 0.0186
2nd Method: P (all 3 red)=
πΆ3
14
πΆ3
50
7. If the probability of winning the race is 5/12,
a) What is the probability of losing the race?
1 β probability of winning
P(Δ) = 1- P(A) = 1 β
5
12
=
7
12
b) What are odds against winning?
π(Δ) =
P(Δ)
P(A)
=
7
12
5
12
β =
7
5
ππ 7: 5
c) If the payoff odd is listed as 6:1, how much profit do you make if you bet $10 and
you win?
Payoff odds against event A = (net profit): (amount bet)
Net Profit = (Payoff odds) Γ (amount bet)
(10 )( 6) = $60
Red:
13/49
Red:
14/50
Blue:
36/50
Red:
12/48
Blue:
34/48
Blue:
35/49
Blue:
36/49
Blue:
36/48
Red:
14/49
Red:
13/48
Red:
13/48
Red:
14/48
Blue:
35/48
Blue:
35/48
6. 6
8. When two different people are randomly selected (from those in your class), find the
indicated probability (assume birthdays occur on the same day of the week with equal
frequencies).
a. Probability that two people are born on the same day of the week.
No particular day is specified, the first person can be born on any day.
π(2ππ ππππ ππ ππππ ππ π‘βπ π πππ πππ¦) =
1
7
π(πππ‘β ππππ πππ ππππ ππ π‘βπ π πππ πππ¦) =
7
7
(
1
7
) =
1
7
b. Probability that two people are both born on Monday.
π(1π π‘ ππππ ππ ππππ ππ πππ) =
1
7
π(2ππ ππππ ππ ππππ ππ πππ) =
1
7
π(πππ‘β ππππ πππ ππππ ππ πππ) =
1
7
(
1
7
) =
1
49
9. How many different auto license plates are possible if the plate has:
Multiplication Counting Rule (The fundamental counting rule):
a) 2 letters followed by 4 numbers?
262
Γ 104
b) 3 letters β no repeats, followed by 3 numbers - repetition allowed?
26 Γ 25 Γ 24 Γ 103
c) 4 letters β repetition allowed, followed by 2 numbers β no repeats?
264
Γ 10 Γ 9
d) 4 places β each character is either a letter or a number?
26 πΏππ‘π‘πππ + 10 π·ππππ‘π = 36 πΆβπππππ‘πππ β
364
7. 7
10. In a first-grade school class, there are 10 girls and 8 boys. In how many ways can:
a. The students finish first, second and third in a foot race? (Assume no ties)
Order is important β permutation is used
ππ
π
= π3
18
= 18 Γ 17 Γ 16
b. The girls finish first and second in a geography contest? (Assume no ties)
Order is important β permutation is used
π
π
π
= π2
10
= 10 Γ 9
c. 3 boys be selected for lunch duty?
Order is not important β combination is used
πΆπ
π
= πΆ3
8
=
8!
3! (5!)
= 56
d. 6 students be selected for a hockey team?
Order is not important β combination is used
πΆπ
π
= πΆ6
18
=
18!
6! (12!)
= 18,564
e. 5 students be selected: 3 boys and 2 girls?
Order is not important β combination is used
πΆ3
8
Γ πΆ2
10
=
8!
3! (5!)
β
10!
2! (8!)
= 2520
f. 4 girls be selected for a field trip?
Order is not important β combination is used
πΆπ
π
= πΆ4
10
=
10!
4! (6!)
= 210
8. 8
Statistics, Sample Test (Exam Review)
Module 2: Chapters 4 & 5 Review
Chapter 5: Discrete Probability Distribution
1. Does the table describe probability distribution? What is the random variable, what are its
possible values, and are its values numerical?
Number of Girls in 3 Births
Number of girls x P(x)
0 0.125
1 0.375
2 0.375
3 0.125
Yes, there are 3 criteria:
1) The values of the random value x (which is the number of girls in three births) are numerical:
0, 1, 2, 3.
2) Their sum: 0.125 + 0.125 + 0.375 + 0.375 =1
3) The values of the random value x are between 0 and 1.
2. In a game, you pay 60 cents to select a 4-digit number. If you win by selecting the correct
4-digit number, you collect $3,000.
a) How many different selections are possible?
n = number of digits = 10: 0, 1, 2β¦,9
10 possible numbers in 4 places (numbers can repeat)
104
= 10,000
b) What is the probability of winning?
P(w) = n(w) / n(S) = 1/10,000
c) If you win, what is your net profit?
net profit = money gained β investment (Cost) = 3000 β 0.60 = $2999.40
d) Write the Probability Distribution of Net Profit if you win.
13. 13
your best prediction about the number of games during the season that had that
many touchdowns scored, round these values to the closest whole number.
π(π₯) =
ππ₯
β πβπ
π₯!
=
5.1055π₯
β πβ5.1055
π₯!
π(0) =
5.10550
β πβ5.1055
0!
= 0.006063
# TD Probability
(0.0001)
Whole Number: Predicted # of Games:
= π·πππ πͺπππππ Γ π΅πππππ ππ πππππ ( ππ πππ)
0 0.006063496 0.006063496(256) = 1.5522 β 2
1 0.03096 0.03096(256) = 7.925 β 8
2 0.079025 0.079025(256) = 20.23 β 20,
3 Continue!
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