Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit A, Slide 1 Probability: Living With The Odds 7

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit A, Slide 1

Probability: Living With The Odds

7


Unit 7A

Fundamentals of Probability


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).


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.


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.


Expressing Probability

The probability of an event, expressed as P(event), is always between 0 and 1 (inclusive).

A probability of 0 means the event is impossible and a probability of 1 means the event is certain.

1

0

0.5

Certain

Likely

Unlikely

50-50 Chance

Impossible

0 ≤ P(A) ≤ 1


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).

=number of ways can occur

( )total number of outcomes

AP A

Theoretical Method forEqually Likely Outcomes


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.

b.

1

4

1

6


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 number of outcomes that are spades 13 1(spade) =

total number of possible outcomes 52 4 P


Outcomes and Events

Scenarios Possible Combinations All 4 Girls {GGGG} 3 Girls and 1 Boy

{GGGB}, {GGBG}, {GBGG}, {BGGG}

2 Girls and 2 Boys

{GGBB}, {GBGB}, {GBBG}, {BGBG}, {BBGG}, {BGGB}

1 Girl and 3 Boys

{GBBB}, {BGBB}, {BBGB}, {BBBG}

All 4 Boys {BBBB}

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?

Of the 16 possible outcomes, 6 have the event two girls and two boys.

P(2 girls) = 6/16 = 0.357


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).

Relative Frequency

number of times occurred( )

total number of observations

AP A


ExampleIf 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?


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.


Three Types of ProbabilitiesA 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.


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


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


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

Probability of an Event Not Occurring


A probability distribution represents theprobabilities of all possible events. To make aprobability 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.

Making a Probability Distribution


All possible outcomes and a probability distribution for the sum when two dice are rolled are shown below.

Possible outcomes

A Probability Distribution


Odds

The odds for an event A are .( )

(not )

P A

P A

The odds against an event A are .(not )

( )

P A

P A

Odds are the ratio of the probability that a particular event will occur to the probability that it will not occur.


Example

What are the odds for getting two heads in tossing two coins? What are the odds against it?

Solution

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.

(2 heads) 1/ 4 1

(not 2 heads) 3 / 4 3

P

P


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.

Documents

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit A, Slide 1 Probability: Living With The Odds 7