Probability of Multiple Events

A marble is picked at random from a bag. Without putting the marble back, a second one has chosen. How does this affect the probability?

A card is picked at random from a deck of cards. Then a die is rolled. How does this affect the probability?

When the outcome of one event affects the outcome of a second event, we say that the events are dependent.

When one outcome of one event does not affect a second event, we say that the events are independent.

Classify each pair of events as dependent or independent.

a. Spin a spinner. Select a marble from a bag that contains marbles of different colors.

Since the two events do not affect each other, they are independent.

b. Select a marble from a bag that contains marbles of two colors. Put the marble aside, and select a second marble from the bag.

Picking the first marble affects the possible outcome of picking the second marble. So the events are dependent.

An expo marker is picked at random from a box and then replaced. A second marker is then grabbed at random.

Two dice are rolled at the same time.

An Ace is picked from a deck of cards. Without replacing it, a Jack is picked from the deck.

Independent

Independent

Dependent

Are the outcomes of each trial dependent or independent events?

1. Roll a number cube. Then spin a spinner

2. Pick one flash card, then another from the stack of 30 flash cards.

6

Independent Events

P(A and B) = P(A) * P(B)

7

If A and B are independent events, then P(A and B) = P(A) * P(B)

Ex: If P(A) = ½ and P(B) = 1/3 then Ex: If P(A) = ½ and P(B) = 1/3 then

P(A and B) =P(A and B) =

6

1

3

1

2

1

A box contains 20 red marbles and 30 blue marbles. A second box contains 10 white marbles and 47 black marbles. If you choose one marble from each box without looking, what is the probability that you get a blue marble and a black marble?

3050

4757

Relate: probability of both events is probability of first event

times probability of second event

Define: Event A = first marble is blue. Then P(A) = .

Event B = second marble is black. Then P(B) = .

Write:  P(A and B) = P(A) • P(B)

The probability that a blue and a black marble will be drawn is , or 49%.4795

At a picnic there are 10 diet drinks and 5 regular drinks. There are also 8 bags of fat-free chips and 12 bags of regular chips. If you grab a drink and a bag of chips without looking, what is the probability that you get a diet drink and fat-free chips?

10

A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.

Solution:The outcome of the coin does not affect the probability of rolling a 6 on the die. These two events are independent.

( 6) ( ) (6)

1 1

2 61

0.08312

P H and P H P

1

2

1

51 1 1

2 5 10

S

T

R

O

P1

2

3

6

5

4

Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel?

P(even) = (3 evens out of 6 outcomes)

(1 vowel out of 5 outcomes)P(vowel) =

P(even, vowel) =

Independent Events

Slide 12

P(jack, factor of 12)

1

5

5

8x =

5

40

1

8

Independent Events

Slide 13

P(6, not 5)1

6

5

6x =

5

36

Independent Events

Slide 14

Multiplication Rule of Probability - Events Involving “And”

11.3 – Events Involving “And”

A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles. What is the probability that a red marble is selected and then a blue one with replacement?

P(Red and Blue) = P(Red) P(Blue)

= 4/9 3/9

= 12/81

= 4/27

= 0.148

Example:

The probability that a particular knee surgery is successful is 0.85. Find the probability that three knee surgeries are successful.

Solution:The probability that each knee surgery is successful is 0.85. The chance for success for one surgery is independent of the chances for the other surgeries.

P(3 surgeries are successful) = (0.85)(0.85)(0.85)

≈ 0.614

Classify each pair of events as independent or dependent.1.A member of the junior class is selected; one of her pets is selected.

2. A member of the junior class is selected as junior class president; a freshman is selected as freshman class president.Use the table shown below to answer the following questions.3. You randomly pick a video and a DVD. What is the probability that you pick an action video and a comedy DVD?

4. What is the probability of randomly picking a drama video and a comedy DVD?

5. Suppose you have seven CDs in a box. Four are rock, one is jazz, and two are country. Today you choose one CD without looking, play it, and put it back in the box. Tomorrow, you do the same thing. What is the probability that youchoose a country CD both days?

17

Movie Collection  Video DVD

Action 12 26

Comedy 14 8

Drama 4 16