ProbabilityChapter 11

1

Odds and Mathematical Expectation

Section 11.6

2

3

Probability to Odds If P(E) is the probability of an event E occurring, then

)not (

)( offavor in Odds

EP

EPE

)(

)not (against Odds

EP

EPE

NOTE: The odds against E can also be found by reversing the ratio representing the odds in favor of E. Also odds is not probability and probability is not odds but one can be found from the other.

ExamplesFind the odds in favor of obtaining

a 2 in one roll of a single die.

an ace when drawing 1 card from an ordinary deck of 52 cards.

at least 1 head when an ordinary coin is tossed 3 times.

4

Solutions

5 to1 5

1

5

6

6

1 offavor in Odds

6561

E

12 to1 12

1

48

4

48

52

52

4 offavor in Odds

5248524

E

1 to7 1

7

1

8

8

7 offavor in Odds

8187

E

5

ExamplesFind the odds against obtaining

a 2 every time in three rolls of a single die.

exactly 2 tails when an ordinary coin is tossed 3 times.

one of the face cards when drawing 1 card from an ordinary deck of 52 cards.

6

Solutions

1 to215 1

215

1

216

216

215against Odds *

2161

216215

E

3 to5 3

5

3

8

8

5against Odds *

8385

E

3 to10 3

10

12

40

12

52

52

40against Odds

52125240

E

7

Examples According to a survey, the probability of being the

victim in a serious crime in your lifetime is 1/20. Find

a.the odds in favor of this event occurring.b.the odds against this event occurring.

19 to1 19

1 offavor in Odds .a

2019201

E

1 to19 against Odds b 119 E.

8

Odds to Probability If the odds in favor of an event E are a to b, then the probability of the event is given by

ba

aEP

)(

9

If the odds in favor of an event E are a to b, then the probability of the event not happening is given by

ba

bEP

)'(

ExamplesThe odds in favor of having complications during surgery in June are 1 to 4.

a.What is the probability that this event will occur?

b.What are the odds against this event occurring?

c.What is the probability that this event will not occur?

10

Solutions

5

1

41

1)( .a

EP

1 to4 ,1

4against Odds .b E

5

4

41

4)not ( .c

EP

11

Expected Value Expected value is a mathematical way to use

probabilities to determine what to expect in various situations over the long run.

It is used to weigh the risks versus the benefits of alternatives in business ventures, and indicate to a player of any game of chance what will happen if the game is played a large number of times.

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Expected Value

If the k possible outcomes of an experiment are assigned the values a1, a2, …, ak and they occur with probabilities p1, p2, …, pk, respectively, then the expected value of the experiment is given by

kk papapaVE 2211)(

13

ExamplesA coin is tossed twice. If exactly 1 head comes up,

we receive $2, and if 2 tails come up, we receive $4; otherwise, we lose $10. What is the expected value of this game?

50.0$10$4$2$)(41

41

42 VE

14

Possible outcomes are HH, HT, TH, and TT.

Examples In a roulette game, the wheel has 38

compartments, 2 of which, the 0 and 00, and the rest are numbered 1 through 36. You can either win $17 if 0 or 00 comes up, or lose $1.00 if any other number comes up. What is the expected value of this game?

05.0$1$17$)(3836

382 VE

15

Examples Suppose you have the choice of selling hot dogs at

two stadium locations. At stadium A you can sell 100 hot dogs for $4 each, or if you lower the price and move to stadium B, you can sell 300 hot dogs at $3 each. The probability of being assigned to A is .55, and to B is .45. Find the expected value for

a. Ab. Bc. Which location would you choose?

16

Solutions

220$)400($55.)1004($55.)( .a A VE

B Stadium .c

405$)900($45.)3003($45.)( .b B VE

17

ExamplesA store specializing in mountain bikes is to open in one

of two malls, described as follows:

1st Mall: Profit if store is successful: $300,000 Loss if it is unsuccessful: $100,000

Probability of success: ½

2nd Mall: Profit if store is successful: $200,000 Loss if it is unsuccessful: $60,000

Probability of success: ¾

Which mall should be chosen in order to maximize the expected profit?

18

Solutions

00,0001$)000,100($)000,300($)mall 1(21

21st E

Mall2 The nd

000,135$)000,60($)000,200($)mall 2(41

43st E

19END

Note: This is ½ because 1 – ½ = ½ not because both parentheses are the same.

Note: 1 – ¾ = ¼ .