Warm-up #1Use the table to answer the questions.

1. What is the probability that someone wearing their seatbelt was going >15 mph over speed limit?

2. What is the probability that someone was going 10-15 mph OR was not wearing a seatbelt?

Number of mph over speed limit

10-15 >15 Total

Wearing Seatbelt

Yes 65 145 210

No 15 75 90

Total 80 220 300

Expected Value

• Essential Question - What is expected value used for in real-life?

What it is and what it is used for

• The expected results of an experiment in the long run.

• Used in business to predict future profits

• Used in insurance to determine how much a person’s insurance rate

• Used in games such as the lottery, slot machines, roulette to determine expected winnings (or losses)

How You Find It

• Probabilities are P

• The values of the events are x

• Expected outcome is E• E = p1x1 + p2x2 +p3x3 …..pnxn

– Multiply each probability by the amount you will win or lose then add them all together.

Interpreting Expected Value

• If you get a ZERO expected value, you expect to BREAK EVEN in the long run

• If you get a POSITIVE expected value, you expect to WIN in the long run

• If you get a NEGATIVE expected value, you expect to LOSE in the long run

• The value you get for expected value will probably NOT be one of the winnings of the game

Example 1

• If you roll a 1, I will give you $4 and if you don’t roll a 1, you give me $1.

• What is the expected value? Does this mean you will win or lose money?

• (prob of 1)($4) + (prob of NOT 1)(-$1)

• You will lose money over time

1 5($4) ( $1)

6 6

0.17

Example 2

You are taking a test that has 4 possible answers for

each question. You get 3 points for each correct answer and lose 1 point for each incorrect answer, and do not gain or lose any points for answers left blank. If you do not know the answer to a question is it to your advantage to guess an answer to a question you don’t know?

Hint:1. Find the probability of each outcome. 2. Find the expected value of guessing the answer

Answer

1. Each question has 4 possible answers; only 1 is correct.

Guessing correctly is ¼ and guessing incorrectly is ¾

2. Multiply the points gained or lost by the correspoding probability. 3 for correct, -1 for incorrect so….

E = 3(1/4) + (-1) (3/4) = 0

¾ - ¾ = 0 so what is your answer? Is it advantageous to guess if you don’t know the answer???

Can we make money?

At a roulette wheel there are 2 zeroes and 36 non zero numbers (18 red and 18 black) to bet on. If I bet $1 on red what is the expected value of my bet?

How about after 10 of the same bets? How much can I be expected to win or lose?

Did I win?

My chance of winning $1 is 18/38 and my chance of losing $1 (or winning -$1) is 20/38.

My expected value, E = (18/38) ( 1 ) + (20/38 )( -1) = 18/38 - 20/38 = -2/38 = -5.26%. I can expect to lose .0526 every spin of the wheelAfter 10 bets I will lose 52.6 = 53 cents.

SpinnerSpinner

What is the expected value of the spinner?What is the expected value of the spinner?

$360018

$450 $1800 $800

$200

$500

$400

$700

$100

$300$600

Tables

x P

3 .25

4 .30

13 .10

2 .35

Use your formula and calculate

(.25) 3 + (.30) 4 + (.10) 13 + (.35) 2 =

.75 + 1.20 + 1.30 + .70 = 3.95

Tables Tables

Find the expected value of Find the expected value of the following event. the following event.

1

173.8

30

6.85

x P

5 .25

8 .30

11 .10

6 .35