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Expectation

Expectation. OBJECTIVE. Find Expected Value of a Distribution. Find the probability of a Binomial distribution. RELEVANCE. Be able to find probabilities of discrete random variables. Definition……. Expected value of a discrete random variable is equal to the mean of the random variable.

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Expectation

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  1. Expectation

  2. OBJECTIVE Find Expected Value of a Distribution. Find the probability of a Binomial distribution.

  3. RELEVANCE Be able to find probabilities of discrete random variables.

  4. Definition…… • Expected value of a discrete random variable is equal to the mean of the random variable. • It plays a role in decision theory (games of chance). • Note: Although probabilities can never be negative, expected value CAN be negative.

  5. Formula…… • Note: It is the same as the formula for the mean.

  6. A Fair Game • In gambling games, an expected value of 0 implies that a game is a fair game (an unlikely occurrence!) • In a profit and loss analysis, an expected value of 0 represents the break-even point. A game is fair if the expectation = 0

  7. 1000 tickets are sold at $1.00 each for a color TV valued at $350. What is the expected value of the gain if a person buys one ticket? Set these up as gains minus losses. Example……

  8. 1000 tickets are sold at $1.00 each for a color TV valued at $350. What is the expected value of the gain if a person buys one ticket? • Answer: • Note: -$0.65 does not mean you lose 65 cents since the person can only win a TV valued at $350. • It means the average of the losses is $0.65 for each of the 1000 ticket holders.

  9. Example • A ski resort loses $70,000 per season when it does NOT snow heavily and makes $250,000 profit when it DOES snow heavily. The probability of having a good season is 40%. Find the expectation of the profit.

  10. A ski resort loses $70,000 per season when it does NOT snow heavily and makes $250,000 profit when it DOES snow heavily. The probability of having a good season is 40%. Find the expectation of the profit. • Answer:

  11. Example • 1000 tickets are sold at $1 each for 4 prizes of $100, $50, $25, and $10. What is the expected value if a person buys 2 tickets?

  12. 1000 tickets are sold at $1 each for 4 prizes of $100, $50, $25, and $10. What is the expected value if a person buys 2 tickets? • Answer:

  13. Example • At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  14. At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain? • Answer: • Because the expected value is negative, you can expect to lose an average of $1.35 for each ticket that you buy.

  15. Binomial Distributions

  16. Many types of probabilities have 2 outcomes. a. A coin: heads or tails b. Baby Born: Male or Female c. T/F Test: True or False • Some situations can be modified or reduced to 2 outcomes. a. A medical treatment: effective or ineffective b. Multiple choice test: correct or incorrect

  17. Binomial Experiment satisfies 4 requirements….. • 1. Has 2 outcomes or reduces to 2. • 2. Fixed # of Trials • 3. Outcomes for each trial must be independent • 4. The probability of success must remain the same for each trial This leads to…

  18. Binomial Distribution….. • Definition – The outcomes of a binomial experiment and their corresponding probabilities.

  19. Notation for the Binomial…… • “n” – number of trials • “p” – the numerical probability of success • “q” – the numerical probability of failure; q = 1 - p • “x” – the # of successes in “n” trials x will always be a whole number – no decimals!

  20. There are many ways to find the probability of a binomial. • One way is to use the formula below:

  21. Example: A coin is tossed 3 times. Find the probability of getting exactly 2 heads. Notations Needed: n = 3 p = ½ q = 1 – ½ = ½ x = 2 Let’s solve the following example using several different methods…..

  22. 1st Way: Using a tree diagram, find the sample space for 3 coins: HHH THH HHT THT HTH TTH HTT TTT There are 3 out of 8 possibilities of getting exactly 2 heads. Probability = 3/8 or 0.375 A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

  23. 2nd way : Using the binomial formula. Remember….. n = 3 p = ½ q = ½ x = 2 A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

  24. 3rd Way: Chart in Book P. 711 in book (or look on copy) Answer: 0.375 A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

  25. 4th and BEST way! – Graphing Calculator Calculator Keys: 2nd Vars 0 or A: binompdf (n,p,x) You will enter binompdf(3, ½, 2) Enter Answer: 0.375 A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

  26. Public Opinion Reported that 5% of Americans are afraid of being alone in the house at night. If a random sample of 20 Americans is selected, find the probability that there are exactly 5 people who are afraid of being alone in the house at night. Answer: n = 20 p = .05 x = 5 Binompdf(20, .05, 5)= 0.002 Example

  27. A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find the probability that exactly 3 will fail. Answer: n = 6 p = 0.05 x = 3 Binompdf(6, 0.05, 3)= 0.002 Example

  28. A student takes a random guess at 5 multiple choice questions. Find the probability that the student gets exactly 3 correct. Each question has 4 possible choices. Answer: n = 5 p = ¼ x = 3 Binompdf(5,1/4,3)= 0.088 Example

  29. Binomial Distributions…Continued At Least And At Most

  30. Public Opinion reported that 5% of Americans are afraid of the dark. If a random sample of 20 is selected, find the probability that A. At most 3 are afraid of the dark. B. At least 3 are afraid of the dark. Example

  31. Public Opinion reported that 5% of Americans are afraid of the dark. If a random sample of 20 is selected, find the probability that…..at most 3 are afraid of the dark. • This is the same as finding the probabilities for x = 0, 1, 2, and 3 • Add the probabilities together. • n = 20 • p = .05 • x = 0, 1, 2, 3

  32. The Long Way……. • Using the calculator: P(at most 3) = bpdf(20, .05, 0) + bpdf(20, .05, 1) + bpdf(20, .05, 2) + bpdf(20, .05, 3) P(at most 3) = 0.358 + 0.377 + 0.189 + 0.060 = 0.984.

  33. Using your graphing calculator: Put x’s in L1 Set a formula for L2. bpdf(20, .05, L1) The sum of L2 is your answer. Is there a shorter way?.......

  34. Public Opinion reported that 5% of Americans are afraid of the dark. If a random sample of 20 is selected, find the probability that…..at least 3 are afraid of the dark. • n = 20 • p = .05 • x = 3, 4, 5, ………, 20

  35. The answer is 0.075.

  36. You Try….. • A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities. a. Fewer than 3 will fail b. None will fail c. More than 3 will fail

  37. n = 6 p = .05 x = 0, 1, 2 The answer is 0.998 A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities.A. Fewer than 3 will fail

  38. n = 6 p = .05 x = 0 The answer is 0.735 You DO NOT need the lists for this one because there is only one x. A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities.B. None will fail

  39. n= 6 p = .05 x = 4, 5, 6 The answer is 0.00008642 A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities.C. More than 3 will fail

  40. Assignment…… • Worksheet

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