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!) In a profit and loss analysis, an expected value of 0 represents the break-even point. A game is fair if the expectation = 0

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

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

1st Way: Using a tree diagram, find the sample space for 3 coins: A coin is tossed 3 times. Find the probability of getting exactly 2 heads. 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

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

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. 3rd Way: Chart in Book P. 711 in book (or look on copy) Answer: 0.375

4th and BEST way! – Graphing Calculator A coin is tossed 3 times. Find the probability of getting exactly 2 heads. 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

Example Answer: n = 20 p = .05 x = 5 0.002 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 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 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.

This is the same as finding the probabilities for x = 0, 1, 2, and 3 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 L1 Set a formula for L2. bpdf(20, .05, L1) The sum of L2 is your answer.

Public Opinion reported that 5% of Americans 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 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 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

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 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 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