+ All Categories
Home > Documents > Probability of Multiple Events (Independent and Dependent Events)

Probability of Multiple Events (Independent and Dependent Events)

Date post: 27-Mar-2015
Category:
Upload: david-ruiz
View: 235 times
Download: 3 times
Share this document with a friend
Popular Tags:
18
Probability of Probability of Multiple Events Multiple Events (Independent and (Independent and Dependent Events) Dependent Events)
Transcript
Page 1: Probability of Multiple Events (Independent and Dependent Events)

Probability of Multiple Probability of Multiple EventsEvents

(Independent and (Independent and Dependent Events)Dependent Events)

Page 2: Probability of Multiple Events (Independent and Dependent Events)

(For help, go to Lesson 1-6.)Warm UpWarm Up

A bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles, and 12 red marbles. Suppose you pick one marble at random. Find each probability.

1. P(yellow) 2.P(not blue) 3.P(green or red)

Page 3: Probability of Multiple Events (Independent and Dependent Events)

Warm Up - SolutionsWarm Up - Solutions

1. total number of marbles = 24 + 22 + 14 + 12 = 72

P(yellow) = = =

2. total number of marbles = 24 + 22 + 14 + 12 = 72

P(not blue) = = = =

3. total number of marbles = 24 + 22 + 14 + 12 = 72

P(green or red) = = = =

1472

72 – 2272

2 • 7 2 • 36

24 + 1272

7 36

5072

3672

2 • 25 2 • 36

2536

1 • 36 2 • 36

12

//

//

Page 4: Probability of Multiple Events (Independent and Dependent Events)

Warm UpWarm Up

Of 300 senior students at Howe High, 150 have taken physics, 192 have taken chemistry, and 30 have taken neither physics nor chemistry. How many students have taken both physics and chemistry?

Page 5: Probability of Multiple Events (Independent and Dependent Events)

Warm Up - SolutionsWarm Up - Solutions

Let x = the number of students who have taken both physics and chemistry.Then (150 – x) is the number of students who have taken physics, but not chemistry. And (192 – x) is the number of students who have taken chemistry, but not physics. 30 students have taken neither, and there are 300 students altogether.

x + (150 – x) + (192 – x) + 30 = 300(150 + 192 + 30) + (1 – 1 – 1)x = 300

372 – x = 300372 – 300 = x

72 = x

So, 72 students have taken both physics and chemistry.

Page 6: Probability of Multiple Events (Independent and Dependent Events)

Consider the Following:Consider the Following:

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

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

Page 7: Probability of Multiple Events (Independent and Dependent Events)

Outcomes of Different EventsOutcomes of Different Events

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

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

Page 8: Probability of Multiple Events (Independent and Dependent Events)

Classify each pair of events as dependent or independent.

Probability of Multiple EventsProbability of Multiple Events

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.

Page 9: Probability of Multiple Events (Independent and Dependent Events)

Decide if the following are Decide if the following are dependent or independentdependent or independent

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

Two dice are rolled at the Two dice are rolled at the same time. same time.

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

Independent

Independent

Dependent

Page 10: Probability of Multiple Events (Independent and Dependent Events)

How to find the Probability of Two How to find the Probability of Two Independent EventsIndependent Events

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

Page 11: Probability of Multiple Events (Independent and Dependent Events)

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?

Let’s Try OneLet’s Try One

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)

P(A and B) = •3050

4757

14102850

= . = Simplify.4795

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

Page 12: Probability of Multiple Events (Independent and Dependent Events)

Mutually Exclusive EventsMutually Exclusive Events

Two events are mutually exclusive Two events are mutually exclusive then they can not happen at the then they can not happen at the same time. same time.

Page 13: Probability of Multiple Events (Independent and Dependent Events)

Probability of Multiple EventsProbability of Multiple EventsAre the events mutually exclusive? Explain.

a. rolling an even number or a prime number on a number cube

By rolling a 2, you can roll an even number and a prime number at the same time.

b. rolling a prime number or a multiple of 6 on a number cube

Since 6 is the only multiple of 6 you can roll at a time and it is not a prime number, the events are mutually exclusive.

So the events are not mutually exclusive.

Page 14: Probability of Multiple Events (Independent and Dependent Events)

How to find the Probability of Two How to find the Probability of Two Mutually Exclusive EventsMutually Exclusive Events

If A and B are mutually exclusive If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)events, then P(A or B) = P(A) + P(B)

If A and B are not mutually exclusive If A and B are not mutually exclusive events, then events, then

P(A or B) = P(A) + P(B) – P(A B)P(A or B) = P(A) + P(B) – P(A B)

Page 15: Probability of Multiple Events (Independent and Dependent Events)

Let’s Try SomeLet’s Try Some

A spinner has ten equal-sized A spinner has ten equal-sized sections labeled 1 to 10. Find the sections labeled 1 to 10. Find the probability of each event. probability of each event.

A)P(even or multiple 0f 5)

B)P(Multiple of 3 or 4)

Hint: Decide if each event is mutually exclusive

No!

P(A)+P(B)-P(A B)

Yes!

P(A)+P(B)

Page 16: Probability of Multiple Events (Independent and Dependent Events)

Let’s Try SomeLet’s Try SomeA)P(even or multiple 0f 5)

B)P(Multiple of 3 or 4)

Page 17: Probability of Multiple Events (Independent and Dependent Events)

Let’s Try OneLet’s Try OneAt a restaurant, customers get to choose one of At a restaurant, customers get to choose one of

four desserts. About 33% of the customers four desserts. About 33% of the customers choose Crème Brule, and about 28% Chocolate choose Crème Brule, and about 28% Chocolate Cheese Cake. Kayla is treating herself for pole Cheese Cake. Kayla is treating herself for pole vaulting four feet at the meet. What is the vaulting four feet at the meet. What is the probability that Kayla will choose Crème Brule probability that Kayla will choose Crème Brule or Chocolate Cheese Cake?or Chocolate Cheese Cake?

Solution:

.33 + .28 = .61 = 61%

Are the events mutually exclusive?Yes. So:

P(A) + P(B)

Page 18: Probability of Multiple Events (Independent and Dependent Events)

Probability of Multiple EventsProbability of Multiple EventsA spinner has twenty equal-size sections numbered from 1 to 20. If you spin the spinner, what is the probability that the number you spin will be a multiple of 2 or a multiple of 3?

P(multiple of 2 or 3) = P (multiple of 2) + P (multiple of 3) – P (multiple of 2 and 3)

= + – 1020

6 20

3 20

= 1320

The probability of spinning a multiple of 2 or 3 is .1320

Are the events mutually exclusive?No. So:

P(A) + P(B) - P(AB)


Recommended