Chapter 4 Probability and Counting Rules Section 4

Chapter 4 Probability and Counting Rules

Section 4 -1 Sample Spaces and Probability

Learning Target Determine sample spaces and find the probability of an event using classical probability or empirical probability.

Basic Concepts Probability Experiment – a chance process that leads to well-defined results called outcomes Outcome – the result of a single trial of a probability experiment Sample Space – the set of all possible outcomes of a probability experiment Examples: tossing a coin (head, tail), roll a die (1, 2, 3, 4, 5, 6), Answer a true/false question (true, false)

Sample Space for Rolling Two Dice Die 2 Die 1 1 2 3 4 5 6 1 (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) 2 (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) 3 (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) 4 (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) 5 (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) 6 (1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6)

Cards in a Regular Deck of Cards 4 suits – spades, diamonds, hearts, clubs 13 of each suit There are 3 face cards in each suit – jack queen, king 52 cards total

Gender of Children Find the sample space for the gender of the children if a family has three children. How can this be done?

Tree diagram Device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment.

Outcomes BBB BBG BGB B BGG 1 st child 2 nd child 3 rd child GBB G GBG GGB GGG

More Vocab Event – consists of a set of outcomes of a probability experiment Simple Event – an event with one outcome Compound Event – event with more than one outcome Example: the event of rolling an odd number on a die

Classical Probability

Probabilities can be expressed as fractions, decimals, or percents. Most problems will be expressed as fractions or decimals. If the problems starts in fractions the answer should be a fraction. If the problem starts as a decimal the answer should be a decimal. Fractions should always be reduced and decimals rounded to two or three decimal places.

Practice Problems A card is drawn at random from an ordinary deck of cards. Find these probabilities. Of getting a jack Of getting a red ace Of getting the 6 of clubs Of getting a 3 or a diamond Of getting a 3 or a 6 If a family has three children, what is the probability that two of the three children are girls?

Solutions

4 Probability Rules

Complementary Events The complement of an event E is the set of all outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by E (read “E bar”) Example: The event E of getting an odd number is 1, 3, 5. The complement of E is getting an even number (2, 4, 6).

Practice Problems Find the complement of each event. Rolling a die and getting a 4 Selecting a letter of the alphabet and getting a vowel Selecting a month and getting a month that begins with a J Selecting a day of the week and getting a weekday

Solutions Getting a 1, 2, 3, 5, 6 Getting a consonant (assume y is a consonant) Getting February, March, April, May, August, September, October, November, or December Getting Saturday or Sunday

Rule for Complementary Events

Empirical Probability

Practice Problems In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. A person has type O blood A person has type A or type B blood A person has neither type A nor type O blood A person does not have type AB blood

Type Frequency A 22 B 5 AB 2 O 21 Total = 50 Solutio n

Law of Large Numbers The larger the number of trials, the closer the empirical probability gets to the classical probability.

Subjective Probability An educated guess regarding the chance that an event will occur

Applying the Concepts and Exercises 4 -1

Section 4 -2 The Addition Rules for Probability

Learning Target IWBAT find the probability of compound events, using the addition rules.

Mutually Exclusive Two events are mutually exclusive if they cannot happen at the same time. In other words they have no outcomes in common. Example: getting a 4 and a 6 are mutually exclusive.

Which ones are mutually exclusive? a. Getting an odd number and getting an even number b. Getting a 3 and getting an odd number c. Getting a 7 and a jack d. Getting a club and getting a king

Addition Rule #1

Practice Problems A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a doughnut at random, find the probability that either is a glazed or chocolate doughnut. At a political rally, there are 20 republicans, 13 democrats, and 6 independents. If a person is selected at random, find the probability that he or she is either a democrat or an independent.

Answers

Addition Rule #2

Practice Problems In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

Answer Staff Females Males Total Nurses 7 1 8 Physicians 3 2 5 Total 10 3 13

For 3 events

Venn Diagrams P(A ) P(B) P(A and B) Mutually Exclusive P(A) Not Mutually Exclusive P(B)

Exercises 4 -2 1 -25 odd and #8

Graded for correct answer 2, 6, 10, 14, 18, 20, 24, 26

Section 4 -3 The Multiplication Rules and Conditional Probability

Learning Target IWBAT find the probability of compound events, using the multiplication rule.

Multiplication Rules The multiplication rules are used to find the probability of events that happen in sequence. For example, when you toss a coin and roll a die, you can find the probability of flipping a head and rolling a 4. The events are independent since the outcome of the first event does not effect the second.

Rule #1

Examples A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen then an ace.

More Examples An urn contains 3 red marbles, 2 blue marbles, and 5 white marbles. A marble is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these. Selecting 2 blue marbles Selecting 1 blue marble then 1 white marble Selecting 1 red marble then 1 white marble

Solutions

Dependent Events When the outcome or occurrence of the first event effects the outcome or the occurrence of the second event in such a way that a probability is changed, the events are said to be dependent. When situations involve not replacing the item that was selected first, the events are dependent.

Conditional Probability The conditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred.

Rule #2 Means that B happens given that A happened first. (Conditional Probability)

Examples Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events. Getting 3 jacks Getting an ace, a king, and a queen in order Getting a club, a spade, and a heart in order Getting 3 clubs

Solutions

Finding Conditional Probability

Example

Solution

Example 2 A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown. Gender Yes No Total Male 32 18 50 Female 8 42 50 Total 40 60 100 Find the probability The respondent answered yes, given that the respondent was female. The respondent was a male, given that the respondent answered no.

Solutions Before you start, determine a variable to represent each outcome. Let M=respondent was male F=respondent was female Y=respondent answered yes N=respondent answered no

a. Respondent answered yes given that the respondent was female.

b. Respondent was male, given the respondent answered no.

Probabilities for “at least” The multiplication rule can be used along with the complement rule to simplify problems involving “at least”.

Example #1

Example #2

Applying the Concepts and Exercises 4 -3

Section 4 -4 Counting Rules

Learning Target IWBAT find the total number of outcomes in a sequence of events, using the fundamental counting rule. IWBAT find the number of ways that r objects can be selected from n objects, using the permutation rule. IWBAT find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule.

The Fundamental Counting Rule

Example Color Red, blue, white, black, green, brown, yellow Type Latex, oil Texture Flat, semigloss, high gloss Use Outdoor, indoor

Factorial Notation

Permutation

Example

Example 2

Combinations

Example

Example 2

Exercises 4 -4 1 -47 odd

Section 4 -5 Probability and Counting Rules

Learning Target IWBAT find the probability of an event using the counting rules.

Example 1

Example 2 A box contains 24 transistors, 4 of which are defective. If 4 are sold at random, find the following probabilities. a. Exactly 2 are defective b. None are defective c. All are defective d. At least one is defective

A. Exactly 2 are defective

B. None are defective

C. All are defective

D. At least 1 is defective

Example 3

Example 4

Example 5

Exercises 4 -5 1 -17 odd Wednesday