Unit 4 – Combinatorics and ProbabilitySection 4.3 – An Introduction to Probability

Calculator Required

Probability = number of possibilities compute first

number of winners compute second

What

How

are

do you w

you do

?

?

in

ing

0 P E 1

P winning 1 P losing

number of winne number ofO dd lor ss s : ser

Ratios reduce like fractions

Independent DependentRoll a Die Twice

6 6Draw two cards with replacement

52 52

Draw two cards without replacement

52

2

Choose two lettersRepetition allowed

26 26

Choose two lettersRepetition NOT allowed

26

2

Consistent DenominatorFor each individual trial

Denominator ReducesFor each individual trial

Find the probability that a four digit number created from thedigits 2, 4, 5, 8 is less than 4000. Assume repetition is notallowed.

4 3 2 1 1 3 2 1

A ball contains three red balls, two blue balls, and one whiteball. If two are drawn and replacement is allowed, find the probability that both are red.

63

63

Two die are rolled. Find the probability that neither is a 5.

65 5

6

Independent vs. Dependent Events Rule of Thumb

-Do the event twice-On the second time of event, check number of possibilities

If the same…..independent…separate fractionsIf different……dependent…...single fraction…. most likely combinations to be used

4 2

1 1

11 11

1 1

6

11

b. A white or red ball is drawn

5 4

1 1

11 11

1 1

9

11

A single ball is drawn from a bag containing four red, five whiteand two green balls. Find the probability of each event. a. A red or green ball is drawn

In a box there are three red, two blue, and three yellow pastels. Doris randomly selects one, returns it, and then selects another.

a. Find the probability that the first pastel is blue and the second pastel is blue 2 2

1 1

8 8

1 1

1

16

b. Find the probability that the first pastel is yellow and the second pastel is red.

3 3

1 1

8 8

1 1

9

64

When Carlos shoots a basketball, the probability that he will make a basket is 0.4. When Brad shoots, the probability of a basket is 0.7. What is the probability that at least one basket is made if Carlos and Brad take one shot each?

P(at least one) = 1 – P(none)

P(at least one basket) = 1 – P(no baskets)

P(Carlos missing) = 0.6 P(Brad missing) = 0.3

1 0.6 0.3

0.82

The probability that Leon will ask Frank to be his tennis partner is ¼, that Paula will ask Frank is 1/3 and that Ray will ask Frank is ¾. Find the probability of each event. a. Paula and Leon ask him.

1 1 1

3 4 12

b. Ray and Paula ask him, but Leon does not

33

4

1 3

4 3 16

The probability that Leon will ask Frank to be his tennis partner is ¼, that Paula will ask Frank is 1/3 and that Ray will ask Frank is ¾. Find the probability of each event. c. At least two of the three ask him.

Leon Yes Paula Yes Ray No

Leon Yes Paula No Ray Yes

Leon No Paula Yes Ray Yes

Leon Yes Paula Yes Ray Yes

11

4

1 1

4 3 48

2

3

1 3 1

4 4 8

1 3 3

3 4 1

3

4 6

1 1 3 1

4 3 4 16

1 1 3 1 19

48 8 16 16 48

The probability that Leon will ask Frank to be his tennis partner is ¼, that Paula will ask Frank is 1/3 and that Ray will ask Frank is ¾. Find the probability of each event.

d. At least one of the three ask him.

P(at least one) = 1 – P(none)

P(at least one will ask) = 1 – P(none ask)

3 2 1

4 3 4

71

8

According to the weather reports, the probability of snow on a certain day is 0.7 in Frankfort and 0.5 in Champaign. Find the probability of each:

7 5 7

10 10 20

7 5 7

10 10 20

3 5 3

10 10 20

a. It will snow in Frankfort, but not in Champaign.

b. It will snow in both cities.

c. It will snow in neither city.

d. It will snow in at least one of the cities.

3 5 171

10 10 20

State the odds of an event occurring given the probability ofthe event.

4a.

9 4

P W9

4 5P L 1

9 9 4 5

ODDS : 4 : 59 9

1b.

12 1

P W12

1 11P L 1

12 12 1 11

ODDS : 1:1112 12

From a standard deck of cards, five are drawn. What are the odds of each selection?

a. five aces

Zero…..there are only four aces in a deck.

b. five face cards

12 40

5 0P W

52

5

12 40

5 0P L 1

52

5

792P W

2598960 2598168

P L2598960

ODDS

792:2598168

33:108257

From a standard deck of cards, five are drawn. What are the odds of each selection?

b. five from one suit

134

5P W

52

5

134

5P L 1

52

5

5148P W

2598960 2593812

P L2598960

ODDS

5148:2593812

2574:129691

From a standard deck of cards, five are drawn. What are the odds of each selection?

b. Two of one suit, three of another

13 1312

3 2P W

52

5

13 1312

3 2P L 1

52

5