Section 4.2Some Probability Rules –

Compound Events

Section 4.2 Some Probability Rules – Compound Events

Learning Targets

After this section, you will be able to:

• Compute the probabilities of general compound events;

• Compute probabilities involving independent events or mutually exclusive events;

• Use survey results to compute conditional probabilities.

Notes from Monday 11/18/13

Review of Sample Space

• An event is any collection of results or outcomes of a procedure.

• A simple event is an outcome or an event that cannot be future broken down into simpler components.

• The 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.

Sample Space ExampleProcedure Example of Event Sample SpaceRoll one die 5 (simple event) {1, 2, 3, 4, 5, 6}

Roll two die 7 (not a simple event)

{1-1, 1-2,…, 6-6}

• When rolling one die, 5 is a simple event because it can’t be broken down further.

• When rolling two dice:• 7 is not a simple event because it can be broken down into

simpler events such as 3-4 or 6-1.• The outcome 3-4 is considered a simple event because it is

an outcome that cannot be broken down further.• 3-4 cannot be broken down into the individual results 3

and 4• 3 and 4 are not individual outcomes when two dice

are rolled.• There are exactly 36 outcomes that are simple events: 1-1,

1-2, …,6-6.

Rolling Dice versus Drawing Cards

Independent Events

Two events are independent if the occurrence

or nonoccurrence of one does not change the

_________________ that the other will occur.

Multiplication Rules of Probability

• Independent Events:

• For any events:

Use when trying to find

the probability of two events

happening together.

Conditional Probability(when )

𝑃 ( 𝐴|𝐵 )= 𝑃 (𝐴𝑎𝑛𝑑 𝐵)𝑃 (𝐵)

Example: Multiplication Rule, Independent Event

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

1s

t

2n

d 1s

t

2n

d 1s

t

2n

d 1s

t

2n

d 1s

t

2n

d 1s

t

2n

d

1 1 2 1 3 1 4 1 5 1 6 1

1 2 2 2 3 2 4 2 5 2 6 2

1 3 2 3 3 3 4 3 5 3 6 3

1 4 2 4 3 4 4 4 5 4 6 4

1 5 2 5 3 5 4 5 5 5 6 5

1 6 2 6 3 6 4 6 5 6 6 6

Example: Multiplication Rule, Dependent Event

Compute the probability of drawing two aces from a well-shuffled deck of 52 cards if the first card is not replaced

before the second card is drawn.

1st card

2nd card

1st card

2nd card

1st card

2nd card

A-H A-D A-H A-C A-H A-S

A-D A-H A-D A-C A-D A-S

A-S A-H A-S A-C A-S A-D

A-C A-H A-C A-D A-C A-S

Practice:Multiplication Rule

Andrew is 55, and the probability that he will be alive in 10 years is 0.72. Ellen is 35, and the probability that she will be alive in 10 years is 0.92. Assuming that the life span of one will have no effect on the life span of the other, what is the probability they will both be alive in 10 years?

1. Are these events dependent or independent?

2. Find P(Andrew alive in 10 years and Ellen alive in 10 years).

Practice: Dependent EventsA quality control procedure for testing Ready-Flash disposable cameras consists of drawing two cameras at random from each lot of 100 without replacing the first camera before drawing the second. If both are defective, the entire lot is rejected. Find the probability that both cameras are defective if the lot contains 10 defective cameras. Since we are drawing cameras at random, assume that each camera in the lot has an equal chance of being drawn.

More than 2 Independent Events

Assume you toss a fair coin, then roll a fair die, and finally draw a card from a standard deck of bridge cards. The three events are independent.

Compute the probability of the outcome heads on the coin and 5 on the die and an ace for the card.

Probability of A or B

The condition of A or B is satisfied by any one of the following conditions:

1. Any outcome in A occurs.

2. Any outcome in B occurs.

3. Any outcome in both A and B occurs.

Practice:Combining Events

Indicate how each of the following pairs of events are combined. Use either the and combination or the or combination.

a. Satisfying the humanities requirement by taking a course in the history of Japan or by taking a course in classical literature.

b. Buying new tires and aligning the tires.

c. Getting an A not only in psychology but also biology.

d. Having at least one of these pets: cat, dog, bird, rabbit.

Examples

1. Compute the probability of drawing either a jack or a king on a single draw from a well-shuffled deck of cards:

2. Compute the probability of drawing a king or a diamond on a single draw:

Mutually Exclusive (disjoint) EventsAddition Rule

• Two events are mutually exclusive or disjoint if they ______________ occur together. In particular, events A and B are mutually exclusive if P(A and B) = ________.

• Addition rule for mutually exclusive events A and B

• Addition rule for any events A and B

Practice:Mutually Exclusive Events

The Cost Less Clothing Store carries seconds in slacks. If you buy a pair of slacks in your regular waist size without trying them on, the probability that the waist will be too tight is 0.30 and the probability that it will be too loose is 0.10.

a. Are the events too tight or too loose mutually exclusive?

b. If you choose a pair of slacks at random in your regular waist size, what is the probability that the waist will be too tight or too loose?

Practice:General Addition Rule

Professor Jackson is in charge of a program to prepare people for a high school equivalency exam. Records show that 80% of the students need work in math, 70% need work in English, and 55% need work in both areas.

a. Are the events needs math and needs English mutually exclusive?

b. Compute the probability that a student selected at random needs math or needs English.

Example:Mutually Exclusive Events

Laura is playing Monopoly. On her next move she needs to throw a sum bigger than 8 on the two dice in order to land on her own property and ass Go. What is the probability that Laura will roll a sub bigger than 8?

Assignment #4.21

Pages 146 – 147

#s 1 - 7Due Tuesday

November 19