Expectation OBJECTIVE Find Expected Value of a Distribution

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. ¡ It plays a role in decision theory (games of chance). ¡ Note: Although probabilities can never be negative, expected value CAN be negative.

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

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

Example…… ¡ 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.

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.

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.

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:

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?

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:

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?

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.

Binomial Distributions

¡ 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

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 The probability of success must remain the same for each trial This leads to… ¡ 4.

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

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!

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

Let’s solve the following example using several different methods…. . ¡ 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

A coin is tossed 3 times. Find the probability of getting exactly 2 heads. ¡ 1 st Way: Using a tree diagram, find the sample space for 3 coins: ¡ There are 3 out of 8 possibilities of getting exactly 2 heads. ¡ HHH HHT HTH HTT ¡ Probability = 3/8 or 0. 375 THH THT TTH TTT

A coin is tossed 3 times. Find the probability of getting exactly 2 heads. ¡ 2 nd 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. ¡ 3 rd 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. ¡ ¡ 4 th and BEST way! – Graphing Calculator Keys: 2 nd Vars 0 or A: binompdf (n, p, x) You will enter binompdf(3, ½, 2) Enter Answer: 0. 375

Example ¡ 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 ¡ A burglar alarm system has 6 failsafe 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 ¡ 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

Binomial Distributions…Continued At Least And At Most

Example ¡ 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.

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 ¡

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.

Is there a shorter way? . . . . ¡ Using your graphing calculator: ¡ Put x’s in L 1 Set a formula for L 2. bpdf(20, . 05, L 1) The sum of L 2 is your answer. ¡ ¡

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 ¡

¡ The answer is 0. 075.

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

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 ¡ 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. B. None will fail ¡ 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. C. More than 3 will fail ¡ n= 6 ¡ p =. 05 ¡ x = 4, 5, 6 ¡ The answer is 0. 00008642

Assignment…… ¡ Worksheet