6 5 Find Expected Value Expected Value A
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6. 5 Find Expected Value
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:
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) + 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 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 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 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 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.