Aim: How are odds at odds with probability and expected value?

Aim: Expectations & Odds Course: Math Lit.

Do Now:

Aim: How are odds at odds with probability and expected value?

Smiles toothpaste is giving away $10,000. All you must do to have a chance to win is send a postcard with your name on it. Is it worthwhile to enter?

depends on number of entries


Expected Value

What if 1,000,000 postcards are received.

Expected Value (your expectation) – of winning contest

Expectation

Amount to Win Probability of Winning

1Expectation $10,000 $0.01

1,000,000

meaning: if you were to play this “game” a large number of times you would expect your average earnings per game to be $0.01.


Expected Value

Fair game – if expected value equals cost of playing the game.

•if expected value is greater than cost (difference between two is positive) then the game is in your favor

•if expected value is less than cost (difference between two is negative) then the game is not in your favor

You draw a card from a deck of cards and are to be paid $10 if it is an ace. What is expected value?

Expectation

Amount to Win Probability of Winning

4Expectation $10 $0.77

52


Compound Expectations

A contest offered one grand prize - $10,000, two 2nd prizes - $5,000 each, and ten 3rd prizes each worth $1,000. What is expected value assuming 1,000,000 entries.

Expectation = Amount to Win Probability of Winning

st 1Expectation 1 $10,000 $0.01

1,000,000

nd 2Expectation 2 $5,000 $0.01

1,000,000

10Expectation 3 $1,000 $0.01

1,000,000rd

Expectation = $0.01 + $0.01 + $0.01 = $0.03


Mathematical Expectation

If an event E has several possible outcomes with probabilities p1, p2, p3, . . . , and if for each of these outcomes the amount that can be won is a1, a2, a3, . . . , respectively, then the mathematical expectation of E is

Expectation = a1p1 + a2p2 + a3p3 + . . .


Model Problem

Two games are offered:

A: Two dice are rolled. You will be paid $3.60 if you roll two ones, and will not receive anything for any other outcome.

B: Two dice are rolled. You will be paid $36.00 if you roll any pair, but you must pay $3.60 for any other outcome.

Which to play?

Expectation = Amount to Win Probability of Winning

1Expectation A $3.60 $0.10

36

minimize losses

maximize winnings


Model Problem

A family has three children. What is the expected number of girls.

B

G

B

G

B

G

B

G

B

G

B

G

B

G

BBB

BBG

BGB

BGG

GBB

GBG

GGB

GGG


Model Problem

A family has three children. What is the expected number of girls.

{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}

# of girls Probability Product

0 1/8 0

1 3/8 3/8

2 3/8 6/8

3 1/8 3/8

Expectation = a1p1 + a2p2 + a3p3 + a3p3

Expectation = 0 + 3/8 + 6/8 + 3/8 = 1.5

meaning: in large number of 3-children families you would expect the average number of girls in a family to be 1.5.


Expectation with Cost

If there is a cost of playing a game, the cost of playing must be subtracted from the expectation.

Expectation = Amt. to Win(Prob. of Winning)

- Cost of Playing

A game consists of drawing a card from a deck of cards. If it is a face card, you win $20. Should you play if it costs $5 to play?

12Expect. = $20 $5 $0.38

52

Winnings Prob. of Winning

Amount to Play


Model Problem

Eva is a realtor and knows that if she takes a listing to sell a house, it will cost her $1000. However, if she sells the house, she will receive 6% of the selling price. If another realtor sells the house, Eva will receive 3% of the selling price. If the house remains unsold after three months, she will lose the listing and receive nothing. Suppose the probabilities for selling a particular $200,000 house are as follows: The probability that Eva will sell the house is 0.4; the probability that another agent will sell the house is 0.2; and the probability that the house will remain unsold is 0.4. What is Eva’s expectation if she takes this listing?


Model Problem

The Payoffs: 6% of 200,000 = $12,000 @ 40%

3% of 200,000 = $6,000 @ 20%

Expectation = Amt. to Win(Prob. of Winning) – Cost of Playing

Expect. = $12000 0.4 $6000 0.2 $1000

= $5,000

Eva sells house

Another agent sells

houseCost of playing


Calculating Odds

For Superbowl 2002, the odds in favor of the St. Louis Rams beating the New England Patriots was 5 to 1.

meaning: if six games were played between the two teams, the Rams would win 5 and the Patriots would win one.

favorable outcome – an experiment that satisfies some event (you set parameters)

unfavorable outcome – an experiment that does not satisfy some event (you set parameters)


Calculating Odds

favorable outcome – an experiment that satisfies some event (you set parameters)

unfavorable outcome – an experiment that does not satisfy some event (you set parameters)

# of favorable outcomesOdds in favor of E =

# of unfavorable outcomes

# of unfavorable outcomesOdds against E =

# of favorable outcomes

Odds of 3/2 are read as “3 to 2”.


Model Problem

A pair of fair dice are rolled once. What are the odds in favor of rolling a sum of 7?

Favorable = 6 Favorable = 30

# of favorable outcomesOdds in favor of E =

# of unfavorable outcomes

6 1Odds in favor of rolling 7 =

30 5


Odds & Probability

The Relationship between Odds & Probability

1. If E is an event in a sample space and the odds in favor of E are a/b, then

2. If E is an event in a sample space, then the

( )a

P Ea b

( )Odds in favor of E

1 ( )

P E

P E


Model Problem

The odds in favor of a particular horse winning a race are 2 to 5. what is the probability that his horse will win the race?

( )a

P Ea b

2 2( )

2 5 7P horse winning


Model Problem

The probability of tossing five coins with a result of three heads and two tails is 5/16, What are the odds in favor of three heads and two tails?

5 5516 16(3 ,2 )

5 11 11116 16

P H T

( )Odds in favor of E

1 ( )

P E

P E

Aim: How are odds at odds with probability and expected value?

Documents

Transcript of Aim: How are odds at odds with probability and expected value?