16.5 Probability of Independent and Dependent

Events -- Working with Conditional Probability

p. 624

• Two events are Independent if the occurrence of 1 has no effect on the occurrence of the other. (a coin toss 2 times, the first toss has no effect on the 2nd toss)

Probability of Two Independent Events

(can be extended to probability of 3 or more Ind. Events)

• A & B are independent events then the probability that both A & B occur is:

• P (A and B) = P (A) • P (B)

Probability of 2 Independent events• You spin a wheel like the one on p. 730.

• During your turn you get to spin the wheel twice. What is the probability that you get more than $500 on your first spin and then go bankrupt on your second spin?

• A = spin > 500 on 1st

• B = bankrupt on 2nd

• P (A and B) = P (A) • P (B) = 8/24 • 2/24 =

• 1/36 = 0.028

Both are ind. events

BASEBALL

• During the 1997 baseball season, the Florida Marlins won 5 out of 7 home games and 3 out of 7 away games against the San Francisco Giants. During the 1997 National League Division Series with the Giants, the Marlins played the first two games at home and the third game away. The Marlins won all three games.

• Estimate the probability of this happening.

• Let A, B, & C be winning the 1st, 2nd, & 3rd games

• The three events are independent and have experimental probabilities based on the regular season games.

• P(A&B&C) = P(A)*P(B)*P(C) =

• 5/7 * 5/7 * 3/7 = 75/343

• =.219

Using a Complement to Find a Probability

• You collect hockey trading cards. For one team there are 25 different cards in the set, and you have all of them except for the starting goalie card. To try and get this card, you buy 8 packs of 5 cards each. All cards in a pack are different and each of the cards is equally likely to be in a given pack.

• Find the probability that you will get at least one starting goalie card.

• In one pack the probability of not getting the starting goalie card is:

• P(no starting goalie) =

• Buying packs of cards are independent events, so the probability of getting at least one starting goalie card in the 8 packs is:

• P(at least one starting goalie) = 1 - P(no starting goalie in any pack)8

• ≈.832

525

524

C

C

PROBABILITIES OF DEPENDENT EVENTS

• Two events A and B are dependent events if the occurrence of one affects the occurrence of the other.

• The probability that B will occur given that A has occurred is called the conditional probability of B given A and is written P(B|A).

Probability of Dependent Events

• If A & B are dependant events, then the probability that both A & B occur is:

• P(A&B) = P(A) * P(B|A)

Finding Conditional Probabilities

The table shows the number of endangered and threatened animal species in the United States as of November 30, 1998. Find

• (a) the probability that a listed animal is a reptile and • (b) the probability that an endangered animal is a

reptile.

Mammals Birds Reptiles Amphibians OtherEndangered 59 75 14 9 198Threatened 8 15 21 7 69

• (a) P(reptile) = number of reptiles = 35 total number of

animals 475

• ≈ .0737

• (b) P(reptile | endangeres) =

• Number of endangered reptiles = 14 total num endangered animals 355

• ≈.0394

Comparing Dependent and Independent Events

You randomly select two cards from a standard 52-card deck. What is the probability that the first card is not a face card (a king, queen, or jack) and the second card is a face card if

• (a) you replace the first card before selecting the second, and

• (b) you do not replace the first card?

• (A) If you replace the first card before selecting the second card, then A and B are independent events. So, the probability is:

• P(A and B) = P(A) • P(B) = 40 * 12 = 30 52 52 169

• ≈ 0.178• (B) If you do not replace the first card before

selecting the second card, then A and B are dependent events. So, the probability is:

• P(A and B) = P(A) • P(B|A) = 40*12 = 40 52 51 221

• ≈ .181

Probability of Three Dependent Events

• You and two friends go to a restaurant and order a sandwich. The menu has 10 types of sandwiches and each of you is equally likely to order any type. What is the probability that each of you orders a different type?

• Let event A be that you order a sandwich, event B be that one friend orders a different type, and event C be that your other friend orders a third type. These events are dependent. So, the probability that each of you orders a different type is:

• P(A and B and C) =

• P(A) • P(B|A) • P(C|A and B)=

• 10/10 * 9/10 * 8/10 =

• 18/25 = .72

A. Draw a tree diagram that shows the given facts.

B. Find the probability of picking a white ball.

C. If a white ball is picked, what is the probability that it came from Jar A?

Jar A contains 1 red, 2 white, and 3 green balls. Jar B contains 4 red and 5 white balls. A die is rolled. If a “1” or a “6” comes up a ball is randomly picked from Jar A. Other wise, a ball is randomly picked from Jar B.

N

N

B

W

A

R G

3

61

6

5

9

1

3

R

2

3

4

9

W

2

6

A.A. Draw a tree diagram that shows the given facts.Draw a tree diagram that shows the given facts.

Find the probability of picking a white ballFind the probability of picking a white ball

1 2 2 5( ) ( ) ( | ) ( ) ( | )

3 6 3 9P W P A P W A P B P W B

13

27

If a white ball is picked, what is the If a white ball is picked, what is the probability that it came from Jar A?probability that it came from Jar A?

)(

) and (|

WP

WAPWAP

11 13 1 279

13 9 27 9 1327

3

13

( ) ( | )

( )

P A P W A

P W

Assignment

P. 626 #1 - 8