Warm UpFind the theoretical probability of each outcome

1. rolling a 6 on a number cube.

2. rolling an odd number on a number cube.

3. flipping two coins and both landing head

up

Find the probability of independent events.

Find the probability of dependent events.

Objectives

Vocabularyindependent eventsdependent events

Adam’s teacher gives the class two list of titles and asks each student to choose two of them to read. Adam can choose one title from each list or two titles from the same list.

Events are independent events if the occurrence of one event does not affect the probability of the other. Events are dependent events if the occurrence of one event does affect the probability of the other.

Classifying Events as Independent or Dependent

Tell whether each set of events is independent or dependent. Explain you answer.

A. You select a card from a standard deck of cards and hold it. A friend selects another card from the same deck.

Dependent; your friend cannot pick the card you picked and has fewer cards to choose from.

B. You flip a coin and it lands heads up. You flip the same coin and it lands heads up again.

Independent; the result of the first toss does not affect the sample space for the second toss.

Try This!

a. A number cube lands showing an odd number. It is rolled a second time and lands showing a 6.

Tell whether each set of events is independent or dependent. Explain you answer.

Independent; the result of rolling the number cube the 1st time does not affect the result of the 2nd roll.

b. One student in your class is chosen for a project. Then another student in the class is chosen.

Dependent; choosing the 1st student leaves fewer students to choose from the 2nd time.

Suppose an experiment involves flipping two fair coins. The sample space of outcomes is shown by the tree diagram. Determine the theoretical probability of both coins landing heads up.

Now look back at the separate theoretical probabilities of each coin landing heads up. The theoretical probability in each case is . The product of these two probabilities is , the same probability shown by the tree diagram.

To determine the probability of two independent events, multiply the probabilities of the two events.

Finding the Probability of Independent Events An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 red marbles and 12 green marbles. What is the probability of selecting a red marble and then a green marble?

Because the first marble is replaced after it is selected, the sample space for each selection is the same. The events are independent.

P(red, green) = P(red) P(green)

The probability of selecting red

is , and the probability of

selecting green is .

Finding the Probability of Independent Events

A coin is flipped 4 times. What is the probability of flipping 4 heads in a row.

Because each flip of the coin has an equal probability of landing heads up, or a tails, the sample space for each flip is the same. The events are independent.

P(h, h, h, h) = P(h) • P(h) • P(h) • P(h)

The probability of landing

heads up is with

each event.

Try This!

An experiment consists of spinning the spinner twice. What is the probability of spinning two odd numbers?

The result of one spin does not affect any following spins. The events are independent.

With 6 numbers on the spinner, 3 of which are odd, the probability of landing on two odd numbers is

P(odd, odd) = P(odd) P(odd)• .

Suppose an experiment involves drawing marbles from a bag. Determine the theoretical probability of drawing a red marble and then drawing a second red marble without replacing the first one.

Probability of drawing a red marble on the first draw

Probability of drawing a red marble on the seconddraw

Suppose an experiment involves drawing marbles from a bag. Determine the theoretical probability of drawing a red marble and then drawing a second red marble without replacing the first one.

To determine the probability of two dependent events, multiply the probability of the first event times the probability of the second event after the first event has occurred.

Application

A snack cart has 6 bags of pretzels and 10 bags of chips. Grant selects a bag at random, and then Iris selects a bag at random. What is the probability that Grant will select a bag of pretzels and Iris will select a bag of chips?

Application Continued

11 Understand the Problem

The answer will be the probability that a bag of chips will be chosen after a bag of pretzels is chosen.

List the important information:

• Grant chooses a bag of pretzels from 6 bags of pretzels and 10 bags of chips.

• Iris chooses a bag of chips from 5 bags of pretzels and 10 bags of chips.

22 Make a Plan

After Grant selects a bag, the sample space changes. So the events are dependent.

Application Continued

After Grant selects a bag, the sample space changes. So the events are dependent.

Draw a diagram.

Grant chooses from: Iris chooses from:

pretzels

chips

Solve33

P(pretzel and chip) = P(pretzel) P(chip after pretzel)•

Grant selects one of 6 bags of pretzels from 16 total bags. Then Iris selects one of 10 bags of chips from 15 total bags.

Application Continued

The probability that Grant selects a bag of pretzels and Iris selects a bag of chips is .

Application Continued

Drawing a diagram helps you see how the sample space changes. This means the events are dependent, so you can use the formula for probability of dependent events.

44 Look Back

Try This!

A bag has 10 red marbles, 12 white marbles, and 8 blue marbles. Two marbles are randomly drawn from the bag. What is the probability of drawing a blue marble and then a red marble?

Try This! Continued

11 Understand the Problem

The answer will be the probability that a red marble will be chosen after a blue marble is chosen.

List the important information:

• A blue marble is chosen from a bag containing 10 red, 12 white, and 8 blue marbles.

• Then a red marble is chosen from a bag of 10 red, 12 white, and 7 blue marbles.

22 Make a Plan

After the first selection, the sample space changes. So the events are dependent.

Draw a diagram.

Try This! Continued

Second choice from:First choice from:

Solve33

P(blue and red) = P(blue) P(red after blue)•

Try This! Continued

One of 8 blue marbles is selected from a total of 30 marbles. Then one of 10 red marbles is selected from the 29 remaining marbles.

The probability that first a blue marble is selected and then a red marble is selected is .

Look Back44

Drawing a diagram helps you see how the sample space changes. This means the events are dependent, so you can use the formula for probability of dependent events.

Try This! Continued

Tell whether each set of events is independent or dependent. Explain your answer.

Lesson Quiz: Part I

1. flipping two different coins and each coin landing showing heads

2. drawing a red card from a standard deck of cards and not replacing it; then drawing a black card from the same deck of cards

Independent; the flip of the first coin does not affect the sample space for the flip of the second coin.

Dependent; there are fewer cards to choose from when drawing the black card.

Lesson Quiz: Part II 3. Eight cards are numbered from 1 to 8 and placed

in a box. One card is selected at random and not replaced. Another card is randomly selected. What is the probability that both cards are greater than 5?

4. An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 yellow marbles and 2 white marbles. What is the probability of selecting a white marble and then a yellow marble?

5. A number cube is rolled two times. What is the probability of rolling an even number first and then a number less than 3?