Chapter 7 - Gordon State College

Report
7
Probability:
Living With
The Odds
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 1
Unit 7A
Fundamentals of
Probability
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 2
Definitions
Outcomes are the most basic possible results of
observations or experiments. For example, if you
toss two coins, one possible outcome is HT and
another possible outcome is TH.
An event consists of one or more outcomes that
share a property of interest. For example, if you toss
two coins and count the number of heads, the
outcomes HT and TH both represent the same
event of 1 head (and 1 tail).
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 3
Example
Consider families of two children. List all the
possible outcomes for the birth order of boys and
girls. If we are only interested in the total number of
boys in the families, what are the possible events?
Solution
There are four different possible birth orders
(outcomes): BB, BG, GB, and GG.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 4
Example (cont)
Because we are asked to consider only the total
number of boys, the possible events with two
children are: 0 boys, 1 boy, and 2 boys. The event 0
boys (0B) consists of the single outcome GG, the
event 1 boy (1B) consists of the outcomes GB and
BG, and the event 2 boys (2B) consists of the single
outcome BB.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 5
Expressing Probability
The probability of an event,
expressed as P(event), is
always between 0 and 1
(inclusive).
0 ≤ P(A) ≤ 1
A probability of 0 means the
event is impossible and a
probability of 1 means the
event is certain.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
1
Certain
Likely
0.5
50-50 Chance
Unlikely
0
Impossible
Chapter 7, Unit A, Slide 6
Theoretical Method for
Equally Likely Outcomes
Step 1:
Count the total number of possible
outcomes.
Step 2:
Among all the possible outcomes, count
the number of ways the event of interest,
A, can occur.
Step 3:
Determine the probability, P(A).
P(A) =
n u m b e r o f w a ys A ca n o ccu r
to ta l n u m b e r o f o u tco m e s
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 7
Example
Apply the theoretical method to find the probability
of:
a. exactly one head when you toss two coins
b. getting a 3 when you roll a 6-sided die
Solution
a. 1
4
b. 1
6
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 8
Example
There are 52 playing cards in a standard deck.
There are four suits, known as hearts, spades,
diamonds, and clubs. Each suit has cards for the
numbers 2 through 10 plus a jack, queen, king, and
ace (for a total of 13 cards in each suit). Notice that
hearts and diamonds are red, while spades and
clubs are black. If you draw one card at random
from a standard deck, what is the probability that it
is a spade?
Solution num ber of outcom es that are spades 13 1

P (spade) =
total num ber of possible outcom es
Copyright © 2015, 2011, 2008 Pearson Education, Inc.

52
4
Chapter 7, Unit A, Slide 9
Outcomes and Events
Assuming equal chance of having a boy or girl at
birth, what is the probability of having two girls and
two boys in a family of four children?
Scenarios
All 4 Girls
3 Girls
and 1 Boy
2 Girls
and 2
Boys
1 Girl and
3 Boys
All 4 Boys
Possible Combinations
{GGGG}
{GGGB}, {GGBG},
{GBGG}, {BGGG}
{GGBB}, {GBGB},
{GBBG}, {BGBG},
{BBGG}, {BGGB}
{GBBB}, {BGBB},
{BBGB}, {BBBG}
{BBBB}
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Of the 16 possible
outcomes, 6 have the
event two girls and two
boys.
P(2 girls) = 6/16 = 0.357
Chapter 7, Unit A, Slide 10
Relative Frequency
A second way to determine probabilities is to
approximate the probability of an event A by making
many observations and counting the number of
times event A occurs. This approach is called the
relative frequency method (or empirical method).
P(A) 
n u m b e r o f tim e s A o ccu rre d
to ta l n u m b e r o f o b se rv a tio n s
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 11
Example
If you are interested only in the number of heads
when tossing two coins, then the possible events
are 0 heads, 1 head, and 2 heads. Suppose you
repeat a two-coin toss 100 times and your results
are as follows:
• 0 heads occurs 22 times.
• 1 head occurs 51 times.
• 2 heads occurs 27 times.
Compare the relative frequency probabilities to the
theoretical probabilities. Do you have reason to
suspect that the coins are unfair?
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 12
Example (cont)
The theoretical probabilities.
0 heads: 1/4, or 0.25
1 head: 2/4 or 0.5
2 heads: 1/4 = 0.25
Find the relative frequencies:
0 heads: 22/100 = 0.22
1 head: 51/100 = 0.51
2 heads: 27/100 = 0.27
The relative and theoretical probabilities are fairly
close. There is no reason to suspect the coins are
unfair.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 13
Three Types of Probabilities
A theoretical probability, based on the assumption
that all outcomes are equally likely, is determined by
dividing the number of ways an event can occur by
the total number of possible outcomes.
A relative frequency probability, based on
observations or experiments, is the relative
frequency of the event of interest.
A subjective probability is an estimate based on
experience or intuition.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 14
Example
Identify the method that resulted in the following
statements.
a. I’m 100% certain that you’ll be happy with this
car.
subjective
b. Based on government data, the chance of dying
in an automobile accident during a one-year period
is about 1 in 8000.
relative frequency
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 15
Example
Identify the method that resulted in the following
statements.
c. The probability of rolling a 7 with a 12-sided die is
1/12.
theoretical probability
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 16
Probability of an Event Not Occurring
If the probability of an event A is P(A), then the
probability that event A does not occur is 1 – P(A).
Since the probability of a family of four children
having two girls and two boys is 0.375, what is the
probability of a family of four children not having
two girls and two boys?
P(not 2 girls) = 1 – 0.375 = 0.625
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 17
Making a Probability Distribution
A probability distribution represents the
probabilities of all possible events. To make a
probability distribution, do the following:
Step 1:
List all possible outcomes. Use a table
or figure if it is helpful.
Step 2:
Identify outcomes that represent the
same event and determine the
probability of each event.
Step 3:
Make a table listing each event and
probability.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 18
A Probability Distribution
All possible outcomes and a probability distribution for
the sum when two dice are rolled are shown below.
Possible outcomes
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Chapter 7, Unit A, Slide 19
Odds
Odds are the ratio of the probability that a particular
event will occur to the probability that it will not
occur.
The odds for an event A are
P(A)
P (n o t A )
The odds against an event A are
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
.
P (n o t A )
P(A)
.
Chapter 7, Unit A, Slide 20
Example
What are the odds for getting two heads in tossing
two coins? What are the odds against it?
Solution
P (2 heads)

P (not 2 heads)
1/ 4
3/4

1
3
The odds for getting two heads in two coin tosses
are 1 to 3. We take the reciprocal of the odds for to
find the odds against, so the odds against getting
two heads in tossing two coins are 3 to 1.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 21
Example
At a horse race, the odds on Blue Moon are given
as 7 to 2. If you bet $10 and Blue Moon wins, how
much will you gain?
Solution
The 7 to 2 odds mean that, for each $2 you bet on
Blue Moon, you gain $7 if you win. A $10 bet is
equivalent to five $2 bets, so you gain 5($7) = $35.
You also get your $10 back, so you will receive $45
when you collect on your winning ticket.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 7, Unit A, Slide 22

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