6 5 Find Expected Value Expected Value A

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6. 5 Find Expected Value

6. 5 Find Expected Value

Expected Value • A collection of outcomes is partitioned into n events, no two

Expected Value • A collection of outcomes is partitioned into n events, no two of which have any outcomes in common. The probabilities of the n events occurring are where. • The values of the n events are • The expected value E of the collection of outcomes is the sum of the products of the events’ probabilities and their values.

Expected Value - Formula • To find expected value we use the formula:

Expected Value - Formula • To find expected value we use the formula:

Example: Find Expected Value • Market research shows that if a bus company adds

Example: Find Expected Value • Market research shows that if a bus company adds a route to Savannah, GA, there is a 36% chance they will make $38, 000. There is a 45% chance they will break even and a 19% chance they will lose $52, 000. Find the expected value of adding a route to Savannah.

 • Use the formula for expected value: E = 0. 36(38, 000) +

• Use the formula for expected value: E = 0. 36(38, 000) + 0. 45(0) + 0. 19(-52, 000) E = $13, 680 + $0 - $9, 880 E = $3800 The expected value to add a route to Savannah, GA is $3800; therefore, it would be advantageous for the company to add the route.

Use Expected Value • A test consists of 23 multiple choice items with 4

Use Expected Value • A test consists of 23 multiple choice items with 4 possible answers for each question. You earn 5 points for each correct answer and lose 3 points for each incorrect answer. If you leave a question blank, no points are added or subtracted from your score. • Is it to your advantage to guess if you do not know the answer?

 • Step 1: Find the probability of each outcome. • There are 4

• Step 1: Find the probability of each outcome. • There are 4 possible answer choices for each question and only 1 correct answer, leaving 3 incorrect answers. The probability of guessing the correct answer is ¼, and the probability of guessing an incorrect answer is ¾. • Step 2: To find the expected value, multiply the points gained or lost by the corresponding probability. • E = 5(1/4) – 3(3/4) = 5/4 – 9/4 = - 4/4 = - 1 • Because the expected value when guessing is -1, it is not to your advantage to guess.

Find Expected Value • A company is having a contest with five prizes. No

Find Expected Value • A company is having a contest with five prizes. No purchase is necessary to win. Prizes, value, and probability of winning each prize are shown in the table. Prize book cd player i-pod Free cell phone for 1 yr. Cash Value $8 $23 $149 $900 P(winning) 0. 0003 0. 00005 0. 000008 0. 00000003 $1500 0. 000002

To find expected value, by finding the sum of the value of each outcome

To find expected value, by finding the sum of the value of each outcome multiplied by its corresponding probability. • 8(0. 0003) + 23(0. 00005) + 149(0. 000008) + 900(0. 00000003) + 1500(0. 000002) = 0. 004745 • The expected value of winning is about $0. 005.