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Mathematics 20
Module 2
Lesson 12
Mathematics 20
Probability
223
Lesson 12
Mathematics 20
224
Assignment 12
Probability
Introduction
Probability can be described as the mathematics of chance. People have always been
interested in predicting or forecasting events and these forecasts are often made on the
basis of remembering what happened in the past. Because a certain event occurred in the
past under certain conditions it is quite likely to happen again under those same
conditions.
There are many games that people play which involve pure chance and winning at these
games cannot be controlled by the player. It is, however, useful for the player to know
what the chance is of winning even though there may be no way of increasing it.
The study of modern probability began in the early seventeenth century when French
mathematicians Fermat and Pascal carried on a correspondence about a gambling
problem. Although gambling and games of chance sparked the study of probability, our
lives are influenced by its application to many other fields. Probability is used in making
weather forecasts, governments and companies use probability in preparing budgets, and
insurance companies set their premiums based on probability.
Experimental probability involves the use of a large amount of past evidence or
observations to make predictions about future events. Weather forecasting and insurance
premiums are prime examples of this. Theoretical probability involves forming a
mathematical model, without experimentation, which can then be used to make
predictions.
In this lesson both types of probability are introduced. Although many informative
examples arise from the use of playing cards and dice, which are often associated with
gambling, these examples will not be used in this lesson since it is easy to fall into the
trap of using only these kinds of examples. In this introduction to probability it is
therefore hoped that the richness and variety of applications of probability to many fields
becomes apparent.
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Assignment 12
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Assignment 12
Objectives
After completing this lesson, you will be able to
•
list the sample space and events for a random experiment.
•
calculate the experimental probability of simple events by performing repeated
experiments.
•
calculate the theoretical probability of an event and the probability of its
complement.
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Assignment 12
Mathematics 20
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Assignment 12
12.1
Introduction to Probability
If a coin is flipped into the air, we are certain that it will come down, but we are not
certain if it will land with heads up or tails up.
However, if we toss that coin many times, (N times) and the number of times it lands with
h
the head side facing up (h times) is recorded, we observe that the ratio
is quite close to
N
1
1
. The more times the coin is tossed, the closer the ratio approaches . In one toss of the
2
2
coin it is impossible to predict if the coin will land with heads up. After several tosses it
becomes possible to say that in approximately one-half of the tosses the coin will land with
heads facing up.
This certainty is the basis of probability since after repetition, order emerges from chaos.
For the coin we can define the probability, or chance, of a head facing up in one toss to be
1
.
2
Probability is a measure of the chance of some event occurring.
Activity 12.11
•
•
•
•
•
Flip a coin 50 times.
Record the number of times the coin lands with head up (h).
h
Calculate the ratio
.
50
Let (t) be the number of times the coin lands with tails facing up.
n
t
t
+
Calculate
and then, calculate
.
50
50
50
Number of Tosses
(N)
10
20
30
40
50
Number of times heads are up (h)
Number of times tails are up (t)
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Assignment 12
A tire manufacturing company makes a quality control test of its tires by selecting a
random sample of 20 tires out of a batch of 1000 tires. These 1000 tires are called the
population, or target group in this test.
The method of selection is that 20 numbers are drawn at random from a box containing
the numbers from 1 to 1000. If number 99 is drawn, then the 99th tire produced is tested
and so on.
Each tire from the sample is then placed on a machine that will simulate normal driving
conditions. The tires are run until they are worn smooth.
Suppose it was found that 1 out of the sample of 20 tires failed during the test while the
other performed well. On the basis of this result it is expected that 50 out of the entire
population of 1000 will fail. The calculation is done by setting up the proportion:
1
x
=
,
20
1000
x = 50
The results of the experiment on the sample space are used to predict the behaviour of the
entire population. The probability of tire failure, on the basis of the experiment, is
1
= 0 .05 .
20
Each of the possible results in an experiment is called an outcome.
An event is a subset of all the possible outcomes.
Activity 12.12
•
Find a styrofoam cup.
•
Toss the cup in the air 50 times.
•
The possible outcomes are:
it will land on the bottom
it will land on the top
it will land on the side
•
Predict what you think the outcome will be.
Was your prediction correct?
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Assignment 12
Example 1
A committee of two persons is to be formed from a group of two men and
three women.
a.
b.
c.
How many different ways are there of choosing this committee?
How many different ways are there of choosing a committee formed by
one man and one woman?
How many different ways are there of choosing a committee formed by
two women?
Solution:
Label the individuals in the group.
a)
Two men - M 1 , M 2 ,
Three women - W1 , W2 , W3
List the set of all possibilities, or outcomes.
M1 , M 2
M 1 , W1
M 1 , W2
M 2 , W2
M 2 , W3
W1 , W2
M 1 , W3
W1 , W3
M 2 , W1
W2 , W3
List the set of the outcomes.
{(M1,M2), (M1,W1), (M1,W2), (M1, W3), (M2, W1), (M2, W2), (M2, W3), (W1, W2),
(W1, W3), (W2, W3)}
•
b)
The event of a committee being formed by two people is the subset of
all of these possible outcomes (10 elements).
List all possibilities, or outcomes.
M1 , M 2
M 1 , W1
M 1 , W2
M 2 , W2
M 2 , W3
W1 , W2
M 1 , W3
W1 , W3
M 2 , W1
W2 , W3
Choose the outcomes with one man and one woman.
List the set of outcomes
M 1 , W1 , M 1 , W2 , M 1 , W3 , M 2 , W1 , M 2 , W2 , M 2 , W3 
•
The event of a committee being formed by one man and one woman contains
six elements.
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Assignment 12
c)
List the set of all outcomes. Choose the outcomes with two women.
W1 , W2 , W1 , W3 , W2 , W3 
•
The event of a committee being formed by two women contains three
elements.
If A is some event in a sample space, then the probability of event A happening is
the number of elements in the event divided by the number of elements in the
sample space.
P  A =
number of elements in the event
number of elements in the sample space
The probability of an event is a number between 0 and 1 that indicates the likelihood that
the event will occur.
•
•
Example 2
An event that is certain to occur has a probability of 1.
An event that is certain not to occur has a probability of 0.
In Example 1, what is the probability of:
a)
the event of a committee being formed from one man and one woman?
b)
the event of a committee being formed from two women?
Solution:
a)
Write the formula.
•
•
P  A =
number of elements in the event
number of elements in the sample space
The number of elements in the sample space: 10 possible outcomes.
The number of elements in the event: 6 ways of forming the committee.
Substitute these values into the formula.
P A  =
6
3
=
10
5
Since there are six out of ten ways of forming a committee of one man and one
woman, we can say that the chance, or probability, of forming such a committee is
6
3
or .
10
5
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Assignment 12
b)
Write the formula.
•
•
P  A =
number of elements in the event
number of elements in the sample space
The number of elements in the sample space: 10 possible outcomes.
The number of elements in the event: 3 ways of forming the committee.
Substitute these values into the formula.
P A  =
The probability of forming a committee of two women is
3
10
3
.
10
Exercise 12.1
1.
Determine the chance (probability) that a kitchen fork, when tossed, will land with
the tines pointing downward.
To do this, make 50 tosses of the same fork and record the number of times it lands
with the tines pointing downward.
What appears to be the probability of the tines pointing downward in a toss of this
particular fork?
2.
Some time prior to spring seeding a farmer wishes to determine how well a
truckload of wheat seed germinates. A random sample of 230 kernels is selected
from the load and is planted. In a few days it was noted that 217 kernels sprouted.
a.
b
c.
d.
e.
f.
What is the population or target group?
What is the sample?
What is the event of importance in this problem and how many elements are
in the event?
What is the probability that a single kernel from the truckload will
germinate?
What percentage of the truckload of seed is expected to germinate?
What is the probability that a single kernel from the truckload will not
germinate?
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3.
In a school cafeteria there is a selection of:
•
•
•
2 deserts
3 main courses
2 salads
d 1 , d 2
m 1 , m 2 ,
s1 , s2
m 3
Each day one student wins a free meal in which one item is selected randomly from
each category of food.
a.
b.
c.
d.
List the set of all possible outcomes.
How many different meals are there in which the main course is m 1 ?
How many different elements are there in the event that the desert is d 1 ?
What is the probability of a meal with s1 being chosen?
12.2 Experimental Probability
A store in Saskatoon wishes to determine if customers prefer scented soap or unscented
soap. A survey is conducted in which the target group, or population, is the entire
population of Saskatoon since the majority of the customers are from the city. It is not
practical to question each individual in the population, so a sample is chosen which is
expected to represent the entire population.
A sample should not be chosen in a haphazard or unplanned way. It is important that it
be chosen in a systematic or random way so as to be a fair picture (representation) of the
entire population. The information gained from the sample can then be used to make
statements and predictions about the entire population. For example, on the basis of the
responses in the sample, the store may decide to stock only one quarter unscented soap
and the remainder scented soap.
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Assignment 12
The following examples illustrate how experimentation and probability can be used to
make predictions.
Example 1
In the manufacture of light bulbs, which are rated as 1000 hour bulbs, it is
necessary that 95% of the bulbs produced actually operate for at least 1000
hours.
In order to ensure the quality of the bulbs each one-hundredth bulb is pulled
off the assembly line and tested.
In a batch of 5000 bulbs the sample of 50 bulbs contained 2 bulbs that failed
before 1000 hours of use.
Are the bulbs achieving the standards that are set?
Solution:
Read the problem.
•
•
•
•
50 bulbs were chosen for the sample.
48 bulbs passed the test and were good bulbs.
What percentage of bulbs were good bulbs?
Is this better than 95%?
Develop a plan.
Let the event of selecting a good bulb be called event A.
This event contains 48 elements out of the 50 in the sample.
If one bulb is to be selected at random from the bath of 5000 bulbs, the probability of
selecting a good bulb is:
P  A =
number of good bulbs
number in sample
Carry out the plan.
P A  =
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number of good bulbs
48
=
= 0 .96
number in sample
50
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Assignment 12
Write a concluding statement.
48
 100 = 96 % of the bulbs passed the test. It can be expected that 96% of
50
the bulbs in the population of 5000 will pass the test.
Therefore,
This is higher than the standard.
Experimental Probability
The probability of an event is:
P  A =
number of favorable outcomes (number in the event)
total number of experiment al trials
NOTE: Probability is always a value between zero and one, it is not a
percent.
Example 2
A bag contains 50 marbles of three different colours:
•
red
•
black
•
white
Determine how many red marbles there are in the bag if the only method
allowed is as follows:
•
Select one marble from the bag at a time without seeing the choice
beforehand.
•
Check its colour.
•
Return it to the bag.
•
Record the data.
Suppose that the following data was obtained when 53 selections were made.
•
•
•
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17 red marbles selected
12 black marbles selected
24 white marbles selected
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Assignment 12
Solution:
Read the problem.
•
•
The sample consists of the 53 selections.
The event of selecting a red marble contains 17 elements.
Develop a plan.
Substitute these values into the formula for determining the probability of an event
occuring.
Carry out the plan.
Write the formula.
P  A =
number of favorable outcomes (number in the event)
total number of experiment al trials
P A  =
Substitute the values into the formula.
17
 0 .32
53
Therefore, the probability of selecting a red marble is 0.32.
Write a concluding statement.
Out of 50 marbles in the bag, it is expected that 50  0 .32 = 16 are red.
The results of this experiment are certainly not exact. Studying a sample which
represents an entire population is a useful way of obtaining information about a
population.
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Assignment 12
Exercise 12.2
1.
Conduct an experiment similar to the one in the previous example using any
collection of 10 similar sized and shaped objects but of three different colours. For
example, you may use objects such as buttons or pieces of paper.
As in the example, you are to select one item at a time , record the colour, and
return the item to the bag. Make about 20 such selections.
a.
b.
c.
d.
2.
On the basis of the sample, predict the number of objects of each colour in the
bag. Check your prediction.
If one selection is made at random from the bag, what is the probability that
it is colour A, colour B, colour C.
What is the sum of the three probabilities?
Repeat the experiment making 40 selections and from the results predict the
number of objects of each colour. Check the prediction.
A survey of 100 people out of a population of 200 000 is to be done to determine how
many watch television at least one time per week during the time period 5 p.m. to 6
p.m.
The results were 74 yes and 26 no.
a.
b.
3.
What is the probability of any person out of the population answering yes to
this question?
How many persons out of the population do not watch television at least once
a week between 5 p.m. and 6 p.m.?
The problem is to predict the area of a certain figure drawn on a sheet of paper.
On an ordinary sheet of paper the size of this page, draw a circle with a diameter
greater than one-third the width of this page. Predict the area of the circle by
following this procedure.
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Assignment 12
Select a small handful of dry beans, (peas, wheat kernels, or something similar).
From about one-half a metre above the page slowly sprinkle the beans on the paper.
Record the number of beans falling inside the circle and the number of beans falling
on the sheet outside the circle. Repeat this several times and complete the given
table.
Trial
Inside Circle
Outside Circle
1
2
3
4
Totals
a.
b.
c.
d.
e.
f.
Determine the probability of a bean landing inside the circle.
Determine the probability of a bean landing outside the circle.
What percentage of the area of the page is the area of the circle?
Measure the page and calculate its area.
What is the predicted area of the circle?
By approximating the diameter calculate the area of the circle. Compare with
the predicted area.
12.3 Theoretical Probability
In section 12.1 it was discussed that when a coin is tossed many times into the air the
probability of the event happening that the coin lands with heads up is:
number of elements in the event
number of experiment al trials
This ratio is usually very close to
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1
if there are an adequate number of tosses.
2
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Assignment 12
Without making any tosses at all we may decide that it is just as likely for the coin to land
with heads up as with tails up since the coin is evenly balanced.
There are only two possible outcomes of the coin landing and the theoretical probability of
the coin landing with heads up is:
number of ways the event can occur
1
=
= 0 .5
number of possible outcomes
2
Therefore, the probability obtained by experiment is a good approximation to the
theoretical probability.
Example 1
In a previous example it was shown how, in a bag of 50 marbles of three
different colours it is possible to predict how many of each colour there are.
If in the bag of 50 marbles, it is known that there are:
•
20 red marbles
•
15 white marbles
•
15 black marbles
What is the probability of selecting, in one draw:
a)
a red marble?
b)
a white marble?
c)
a black marble?
Solution:
a)
red marble:
b)
white marble:
c)
black marble:
number of
number
number of
number
number of
number
ways the event can occur
20
=
= 0 .4
of possible outcomes
50
ways the event can occur
15
=
= 0 .3
of possible outcomes
50
ways the event can occur
15
=
= 0 .3
of possible outcomes
50
Therefore the probability of choosing a red marble is
20
= 0 .4 , a white marble is
50
15
15
= 0 .3 , and a black marble is
= 0 .3
50
50
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Assignment 12
Theoretical Probability
If A is the desired event, the theoretical probability of A occurring is:
P  A =
Example 2
number of outcomes in the event
total number of possible outcomes
Two coins are tossed in the air. What is the probability that each lands with
heads up?
Solution:
List all the possible outcomes.
•
•
•
If the first coin comes up heads, the other can come up heads or tails to give
{HH, HT}.
If the first coin comes up tails, the other coin can come up heads or tails to give
{TH, TT}.
The total list of outcomes is {HH, HT, TH, TT}.
Determine how many outcomes there are when the coins both land heads up.
•
There is only one outcome, (HH), that both coins come up heads.
Determine the probability that the coins both land heads up.
•
The probability of two heads coming up is:
P A  =
Example 3
number of outcomes in the event
total number of possible outcomes
=
1
4
Three coins (a quarter, a dime and a nickel) are tossed in the air. What is the
probability that each lands with heads up?
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Assignment 12
Solution:
A tree diagram is used to help list all possible outcomes when three coins are tossed.
Quarter
Dime
H
Nickel
H
T
H
T
H
T
H
H
T
T
T
H
T
List all the possible outcomes.
•
There are eight different paths as indicated by the branches of the tree and the
paths are used to list all possible outcomes.
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Determine how many outcomes that all three coins land heads up.
•
Within the list of possibilities there is only one way {HHH} all coins come up all
heads.
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Assignment 12
Determine the probability that all three coins land heads up.
•
The probability of the event of all heads coming up is:
P A  =
number of outcomes in the event
total number of possible outcomes
=
1
8
If A is any event whose probability of happening is p , then the event of A not happening
is called the complement of A, written A , and its probability is 1  p .
 
P A  + P A = 1
Example 4
A bag contains 10 identically shaped blocks, 6 of which are blue and the
remainder green. The bag is well shaken and one block is drawn at random
(without looking inside the bag).
a.
b.
c.
d.
e.
What is the probability that the block is green?
What is the probability that the block is not green?
What is the probability that the block is either blue or green?
What is the probability that the block is neither blue nor green?
If one blue block is drawn and not replaced, what is the probability
that the next block drawn is blue?
Solution:
a.
b.
c.
d.
e.
P  A = P ( block is green ) =
4
= 0 .4
10
6
P ( A ) = P block is green =
= 0 .6
10
10
P ( block is blue or green ) =
= 1
10
0
P block is blue or green =
= 0
10
Five blue blocks and four green blocks remain.
5
P ( blue block on second draw ) =
9


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

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Assignment 12
Exercise 12.3
1.
A spinner is divided into seven equal sectors numbered from 1 to 7.
a.
b.
c.
Assuming that the spinner is fair, what is the theoretical probability that the
arrow will stop on the sector numbered 6?
What is the theoretical probability that the arrow will stop on an even
numbered sector?
What is the probability of the complement of the event of the arrow stopping
on a 2 or a 3?
2.
A spinner is divided into 4 sectors of unequal area. Describe an experiment that
would allow you to estimate the probability of the arrow stopping on each of the
four sectors.
3.
A hat contains two red and two blue tickets. Three tickets are selected at random,
one at a time without replacement.
a.
b.
c.
d.
e.
Draw a tree diagram to illustrate all the possible situations.
List all the possible outcomes.
List all the possible outcomes in the event that one ticket is red.
List all the possible outcomes in the event that at least one ticket is red.
What is the probability of the event that two tickets are not blue?
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Assignment 12
4.
Suppose a hat contains three red tickets and three blue tickets. Three tickets are
selected at random, one at a time without replacement.
a.
b.
5.
Draw a tree diagram to illustrate all the possible situations.
What is the probability of the event of drawing all three tickets of the same
colour?
A box contains equal sized and shaped blocks numbered from 1 to 4. Three blocks
are drawn, one at a time without replacement.
a.
b.
c.
Draw a tree diagram to illustrate all possible cases.
What is the probability of drawing three consecutive numbers in any order?
What is the probability of the complement of the event of drawing three
numbers whose sum is 4?
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Assignment 12
Summary
The concepts that you have learned in this lesson are:
•
It is necessary to know the following definitions of terms:
Population
(Target Group) This is the entire set of objects being studied.
Sample
(Sample Space or Random Sample) This is the subset of the
population which is expected to be a fair representation of the
entire population.
Outcomes
The set of all possibilities of a certain situation.
Event
Probability
This is a subset of a set of outcomes.
This is a measure of the chance of some event occurring. It is
defined by this formula
The probability of event A happening is
P(A) =
Complement
Mathematics 20
number of elements in the event
number of elements in the sample space
If A is any event whose probability of happening is p , then the
event of A not happening is called the complement of A, written
A , and its probability is 1  p .
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Assignment 12
Answers to Exercises
Exercise 12.1
Exercise 12.2
1.
2.
Answers will vary.
a.
The truckload of wheat seed is the target group.
b.
230 kernels
c.
The event of importance is that a kernel sprouts. There
are 217 kernels (elements) in the event.
217
P( A ) 
 0 .94
d.
230
e.
94%
1  0 .94  0 .06
f.
3.
a. {d 1 m 1 s1 , d 1 m 1 s2 , d 1 m 2 s1 , d 1 m 2 s2 , d 1 m 3 s1 , d 1 m 3 s2 ,
d 2 m 1 s1, d 2 m 1 s2 , d 2 m 2 s1, d 2 m 2 s2 , d 2 m 3 s1, d 2 m 3 s2 }
b. 4
c. 6
6
1
=
d.
12
2
1.
Answers will vary.
2.
a.
74
 0 .74
100
Therefore the probability of answering yes is 74%
200 000 × 0.74 = 148 000
Out of a population of 200 000 it is expected 148 000
watch television at least once per week between 5 p.m.
and 6 p.m.
b.
200 000 × 0.26 = 52 000
Out of a population of 200 000, 52 000 do not watch
television once a week between 5 p.m. and 6 p.m.
Exercise 12.3
1.
a.
b.
c.
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1
7
3
7
5
7
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Assignment 12
2.
Possible solution:
•
•
•
3.
Mathematics 20
Spin the arrow 100 times.
Record the number of times the arrow stops on each
section.
The probability of landing on the larger sector should be
greater than the probability of landing on the smaller
sector.
a.
b.
{RRB, RBB, RBR, BRB, BRR, BBR}
c.
{RBB, BRB, BBR}
d.
{RRB, RBB, RBR, BRB, BRR, BBR}
e.
3
1
=
6
2
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Assignment 12
4.
a.
R
R
R
B
R
B
B
R
R
B
B
B
R
B
b.
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2 1

8 4
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Assignment 12
5.
a.
b.
c.
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12
1
=
24
2
24
1
24
250
Assignment 12
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Module 2
Assignment 12
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Assignment 12
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Assignment 12
Optional insert: Assignment #12 frontal sheet here.
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Assignment 12
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Assignment 12
Assignment 12
Values
(40)
A.
Multiple Choice: Select the best answer for each of the following and place a
() beside it.
1.
Sluggo's batting average over the last 50 games was a steady 0.4375,
because the statistics showed that for the last 400 times at bat he got
on base with a hit 175 times. The probability that Sluggo will hit the
ball and get on base the next time at bat is ***.
____
____
____
____
2.
a.
b.
c.
d.
0
0.035
0.965
1
If the probability of event A is 0.04 and there are 52 elements in the
event, then the number of elements in the sample space is ***.
____
____
____
____
Mathematics 20
0.4375
0.175
437.5
4375
If the probability of a tornado occurring in Saskatchewan this summer
is 0.035, then the probability of a tornado not occurring is ***.
____
____
____
____
3.
a.
b.
c.
d.
a.
b.
c.
d.
1300
208
52
2.08
255
Assignment 12
4.
5.
6.
A bag contains 2 red, 3 green, and 6 white marbles. The probability of
selecting either a red or a white marble in a random draw is ***.
____
a.
____
b.
____
c.
____
d.
A bag contains red and green marbles. If the probability of drawing a
16
red marble is
, the probability of drawing a green marble is ***.
25
____
a.
____
b.
____
c.
____
d.
9
25
16
25
9
16
11
25
A bag contains red and green marbles. If the probability of drawing a
3
red marble is , and there are 126 red marbles, the number of green
5
marbles is ***.
____
____
____
____
Mathematics 20
8
12
6
11
2
11
8
11
a.
b.
c.
d.
84
85
200
210
256
Assignment 12
7.
A letter is drawn at random from the letters in the word
P R O B A B I L I T Y. The probability that it is a vowel is ***.
____
a.
____
b.
____
c.
____
d.
1
11
1
4
3
11
5
11
For questions 8 to 12 use the following situation.
A box contains 20 slips of paper numbered 1 to 20. A slip of paper is drawn
at random from the box without replacement and then another slip of paper
is drawn.
8.
9.
The probability of the event that an even number is drawn in the first
draw is ***.
____
a.
____
____
____
b.
c.
d.
The number of elements in the event of drawing a number not less
than 8 in the first draw is ***.
____
____
____
____
Mathematics 20
1
20
0.5
0.4
10
a.
b.
c.
d.
7
8
12
13
257
Assignment 12
10.
11.
12.
13.
The probability of drawing a number divisible by 3 in the first draw is
***.
____
a.
____
b.
____
____
c.
d.
The probability of drawing an odd number in the second draw after an
8 has been drawn in the first draw is ***.
____
a.
____
b.
____
c.
____
d.
1
2
9
20
10
19
9
19
The probability of drawing a prime number in the first draw is ***.
____
a.
____
b.
____
c.
____
d.
1
3
2
5
9
20
8
If the germination in a small sample of tomato seeds is 95%, the
probability of a seed selected from the target group not germinating is
***.
____
____
____
____
Mathematics 20
0.3
1
3
1
6
a.
b.
c.
d.
0.95
0.05
5%
0.5
258
Assignment 12
14.
15.
16.
Mathematics 20
2
The expression  
3
____
a.
____
b.
____
c.
____
d.
3
2
 3 a 2 b 1 

 is equivalent to ***.
2 
  8a 
8
a
24 b2
1
24 b2
16
a
24 b2
8
a b
24
The expression
2x  5 
is not defined at ***.
2 x  x  15
2
____
a.
5
____
b.
____
c.
____
d.
5
, 5
2
5
3,  , 5
2
5
3, 
2
 3,
The lowest common denominator for
____
____
a.
b.
____
____
c.
d.
1
3
is ***.
+ 2
x  1
x + 1
2
x
 1x + 1
either x  1 or x  1
( x  1)( x + 1)2
x  1x + 1x2 + 1
259
Assignment 12
17.
18.
19.
20.
Mathematics 20
The equivalent expression to
____
a.
____
b.
____
c.
____
d.
3x + 1
x

is ***.
xx  2 
2  x
3 x + 1  x2
xx  2 
2
x + 3x + 1
xx  2 
2x + 1
x  x  2 2  x 
3 x2 + x
xx  2 
2
 2 
The expression  1 
x 
4
____ a.
x2
x2
____ b.
4
1
____ c.
4 x2
4
____ d.
x
2
in simplified form is ***.
By long division 3 x3 + 5 x  1  x + 2  is equivalent to ***.
____
a.
3 x 3  6 x  17 
____
____
b.
c.
3 x 3  6 x  17
3x3 1
____
d.
3 x 3  6 x  17 
By long division
____
a.
____
____
____
b.
c.
d.
35
x2
34
x2
2x + 5
is equivalent to ***.
x  3
11
x  3
x  2
x  3
2 x + 5 x  3 
2+
260
Assignment 12
Part B can be answered in the space provided. You also have the option to do
the remaining questions in this assignment on separate lined paper. If you
choose this option, please complete all of the questions on the separate paper.
Evaluation of your solution to each problem will be based on the following.
(40) B.
(8 each)
•
A correct mathematical method for solving the problem is shown.
•
The final answer is accurate and a check of the answer is shown where
asked for by the question.
•
The solution is written in a style that is clear, logical, well organized,
uses proper terms, and states a conclusion.
1.
A manufacturer of batteries tests a sample of 250 batteries selected
from a lot of 30 000 batteries. The test was run to determine whether
the batteries last the advertised life span of 50 hours. It was found
that 238 batteries were still working after the 50 hours.
Mathematics 20
a.
What is the probability that a battery selected at random from
the remaining lot will run at least 50 hours?
b.
If a store takes a shipment of 300 batteries, how many in the
shipment would be reasonably expected to fail to run the
advertised life span of 50 hours.
261
Assignment 12
2.
A committee of three persons is to be chosen from a group of three men
and two women to fill the position of president, secretary and
treasurer.
a.
Mathematics 20
Draw a tree diagram which illustrates all the possible outcomes.
262
Assignment 12
3.
Mathematics 20
b.
List all the elements in the event that the committee consists of
two women.
c.
What is the probability exactly one man is on the committee?
d.
What is the probability that no women are on the committee?
From the set of numbers  3,  2,  1, 1, 2 a number is selected at
random and is not replaced. Then another number is selected at
random. Let the first number represent the x-coordinate and the
second number the y-coordinate.
a.
List all the possible outcomes.
b.
What is the probability that the ordered pair belongs to the
third quadrant?
c.
What is the probability that the ordered pair does not belong to
the first quadrant?
263
Assignment 12
4.
Mathematics 20
In a bag there are 4 red, 3 white, and 2 black marbles. One marble is
picked from the bag at random and replaced. Find the probability of
each of the following.
a.
picking a red marble
b.
picking a white marble
c.
picking a black marble
d.
picking either a red or a white marble
e.
picking neither a red nor a black marble
f.
picking a blue marble
264
Assignment 12
5.
Draw a tree diagram to illustrate all possible outcomes of names for a
child if there are four choices for the first name and three choices for
the middle name.
Answer Part C on separate lined paper. Please include any tables or graphs that
you are required to do with the assignment.
(20) C.
(10 each)
1.
Use the diagram on the following page to answer this question.
a.
b.
2.
Suppose that two boys and two girls are to be arranged in a row.
a.
b.
c.
Mathematics 20
Write a procedure to determine the ratio of area covered by the
closed curve compared to the area of the entire page.
Carry out your procedure and determine this ratio. Express your
answer as a percentage.
Draw a tree diagram illustrating all possible arrangements.
If the arrangement is done at random, what is the probability of
having two girls next to each other?
What is the probability that both boys are in the middle of the
row?
265
Assignment 12
Mathematics 20
266
Assignment 12