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Chapter 4Probability

4-1 Overview

4-2 Fundamentals

4-3 Addition Rule

4-4 Multiplication Rule: Basics

4-5 Multiplication Rule: Complements and Conditional Probability

4-6 Probabilities Through Simulations

4-7 Counting

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Section 4-1Overview

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Overview

Rare Event Rule for Inferential Statistics:

If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct.

Statisticians use the rare event rule for inferential statistics.

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Example

• We have a gender selection procedure that claims to increase the birth of baby girls.

• Actual results out of 100 babies born: 98 girls, 2 boys.

• Is there a chance of getting boys?

• What’s the probability?

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Section 4-2 Fundamentals

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Key Concept

This section introduces the basic concept of the probability of an event. Three different methods for finding probability values will be presented.

The most important objective of this section is to learn how to interpret probability values.

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Example

• We have a quality control test that shows that there are 5 defective printers and 15 that are “good.”

• What is the probability of choosing one that is not defective?

• What is the probability of choosing one that is defective?

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Definitions

Event

any collection of results or outcomes of a procedure

Simple Event

an outcome or an event that cannot be further broken down into simpler components

Sample Space

for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any

further

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Example

• In the following display, we use f to denote a female baby and m to denote a male baby.

Procedure Example of event

Single birth female

(simple event)

3 births 2 females and 1 male

(not a simple event)

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Example

• In the following display, we use f to denote a female baby and m to denote a male baby.

Procedure Example of event Sample SpaceSingle birth female {f, m}

(simple event)3 births 2 females and 1 male {fff, ffm, fmf,

(not a simple event) fmm, mff, mfm, mmf, mmm}

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Notation for Probabilities

P - denotes a probability.

A, B, and C - denote specific events.

P (A) - denotes the probability of event A occurring.

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Basic Rules for Computing Probability

Rule 1: Relative Frequency Approximation of Probability

Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P(A) is estimated as follows:

P(A) = number of times A occurred

number of times trial was repeated

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Example

Find the probability that NBA basketball player Reggie Miller makes a free throw after being fouled. At one point in his career, he made 5915 free throws in 6679 attempts (NBA data).

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Activity

• Flip a penny 30 times. Record your results on the board for # of heads flipped. Let A = # of heads flipped.

Trial # Heads Tails P(A)

• Plot your data (x-axis: trial number, y-axis:

proportion)

What do you notice about the graph?

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Law of Large Numbers

As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability.

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Basic Rules for Computing Probability - cont

Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes)

Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then

P(A) = number of ways A can occur

number of different simple events

sn =

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Example

1) If you have four kings from a deck, what’s the probability you’ll get a king of diamonds?

2) If trying to determine P(2) with a fair and balanced die, each of the six faces has an equal chance of occurring. Find P(2).

3) Find the probability that when a couple has 3 children, they will have exactly 2 boys. Assume that boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child.

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Basic Rules for Computing Probability - cont

Rule 3: Subjective Probabilities

P(A), the probability of event A, is estimated

by using knowledge of the relevant

circumstances.

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Examples

1) When trying to estimate the probability of

rain tomorrow, meteorologists use their

expert knowledge of weather conditions to

develop an estimate of the probability.

2) What is the probability that your car will

be hit by a meteorite this year?

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Probabilities/Outcomes that aren’t equally likely

• Is the probability that a Republican will win the next presidential election 1/2 ? In other words, is the probability that a Republican will win equal to the probability that a Democrat will win?

• Is there a ½ probability that you will pass your next statistics test?

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Warning!

When you know nothing about the likelihood of different possible outcomes, don’t assume they are equally likely!

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Probability Limits

The probability of an event that is certain to occur is 1.

The probability of an impossible event is 0.

For any event A, the probability of A is between 0 and 1 inclusive. That is, 0 P(A) 1.

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Possible Values for Probabilities

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Example

• If a year is selected at random, find the probability that Thanksgiving Day will be on a:

a) Wednesday

b) Thursday

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DefinitionThe complement of event A, denoted by

A, consists of all outcomes in which the

event A does not occur.

Example: In reality, more boys are born

than girls. In one typical group, there are

205 newborn babies, 105 of whom are

boys. If one baby is randomly selected

from the group, what is the probability

that the baby is not a boy?

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Rounding Off Probabilities

When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When the probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.)

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Definitions

The actual odds against event A occurring are the ratio P(A)/P(A), usually expressed in the form of a:b (or “a to b”), where a and b are integers having no common factors.

The actual odds in favor of event A occurring are the reciprocal of the actual odds against the event. If the odds against A are a:b, then the odds in favor of A are b:a.

The payoff odds against event A represent the ratio of the net profit (if you win) to the amount bet.

payoff odds against event A = (net profit) : (amount bet)

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Odds! (Vegas, baby!)If you bet $5 on the number 13 in roulette, your

probability of winning is 1/38 and the payoff odds are given by the casino as 35:1.

a) Find the actual odds against the outcome of 13.

b) How much net profit would you make if you win by betting on 13?

c) If the casino were operating “just for the fun of it” (yeah, right), and the payoff odds were changed to match the actual odds against 13, how much would you win if the outcome were 13?

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Recap

In this section we have discussed:

Rare event rule for inferential statistics.

Probability rules.

Law of large numbers.

Complementary events.

Rounding off probabilities.

Odds.