4.2 SOME PROBABILITY RULES – COMPOUND EVENTS

PART 1: MULTIPLICATION RULES

Chapter 4: Elementary Probability Theory

Definitions

A compound event consists of two or more simple events.

Two events are independent if the occurrence or non-occurrence of one event does not change the probability that the other event will occur.

If events are dependent, the probability of one event depends upon the occurrence of the other event.

The type of events determines the way we compute the probability of the two events happening together.

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Multiplication Rules

The event A and B consists of all experimental outcomes that are in both of the events A and B. This is called the intersection and uses the symbol .

Multiplication Rule for Independent Events

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𝑃 ( 𝐴𝑎𝑛𝑑𝐵 )=𝑃 ( 𝐴⋂𝐵 )=𝑃 ( 𝐴 ) ∙𝑃 (𝐵 )

The Event A and B

Figure 4-4(a)

Conditional Probability

If the events are dependent, then we must take intoaccount the changes in the probability of one event caused by the occurrence of the other event.

Conditional probability is the probability that a dependent event will occur given that another event has occurred.

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Independence

Recall: Two events are independent if the occurrence or non-occurrence of one event does not change the probability that the other event will occur.

Two events A and B are said to be independent if Because the Probability of Event B did/does

not affect the probability of A

General Multiplication Rule for ANY Events

Use either, depending of the information Note that the conditional probability rule is

contained in both formulas

Multiplication RulesPage 143

How to Use the Multiplication Rules

1. Determine whether A and B are independent events or dependent events.

2. If A and B are independent eventsP(A and B) = P(A)•P(B)

3. If A and B are any eventsP(A and B) = P(A)•P(B|A)

or P(A and B) = P(B)•P(A|B)

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The Multiplication Rules apply whenever we want to determine the probability of two events happening together. To indicate “together” we use and between the events. (Page 145)

The multiplication rule for independent events can be expanded for more than two independent events… keep multiplying!

Example 4 – Mult. Rule, Independent Events

Suppose you are going to throw two fair dice. What is the probability of getting a 5 on each die?

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P(5 on 1st die and 5 on 2nd die) = P(5 on 1st die) • P(5 on 2nd die)

Example 4 – Mult. Rule, Independent Events

Suppose you are going to throw two fair dice. What is the probability of getting a 5 on each die? (use the sample space)

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1. Write out the sample space

2. What is the total number of outcomes?

3. How many “favorable” outcomes are there?

Sample Space for Two Dice

Figure 4-2

Consider a collection of 6 balls that are identical except in color. There are 3 green balls, 2 blue balls, and 1 red ball. Compute the probability of drawing 2 green balls from the collection if the first ball is not replaced before the second ball is drawn.

Example 4 – Mult. Rule, Dependent EventsPage 145

Solution – Mult. Rule, Dependent Events

P( green ball 1st draw and green ball 2nd draw) = P(green on 1st) • P(green on 2nd | green on first)

2: because assuming we got a green ball on 1st draw, there are only 2 left.5: because if you remove a ball and do not replace, there are only 5 left.

Assignment

Page 155 #2, 5, 6, 12, 15, 19, 21, 27(not part f)