HSS.MD.B.5Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.MD.B.5: Expected Payoffs, Games of Chance and Comparing Strategies
In plain English: HSS.MD.B.5 is an advanced (+) Common Core statistics and probability standard that asks students to weigh the outcomes of a decision by assigning probabilities to payoff values and finding expected values. Students find the expected payoff of a game of chance, such as a lottery ticket, and compare strategies, such as two car insurance policies. It is usually taught in a statistics course or Precalculus.
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a.Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
b.Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
Common Core State Standards for Mathematics · Domain: Using Probability to Make Decisions (MD) · Cluster: Use probability to evaluate outcomes of decisions Also written as HSS-MD.B.5 or S-MD.5 · Official standard
Students use expected value to make decisions. They list the outcomes of a choice, assign a probability and a payoff (a gain or a cost) to each, and compute the expected payoff. Standard a applies this to games of chance, such as a scratch-off ticket or a fast-food game piece; standard b uses it to compare strategies, such as a high-deductible and a low-deductible car insurance policy.
The lesson also treats expected value as one input to a decision, not the whole decision. Students see that a choice with a better expected value can carry more risk, and they explain in context when a person might reasonably choose differently.
Learning Objectives
By the end of this lesson, students will be able to:
Assign probabilities and payoff values, including the cost to play, to every outcome of a game or decision
Find the expected payoff (expected net winnings) of a game of chance and decide whether the game is fair
Compare two or more strategies by their expected values and choose the better one for a stated goal
Compare a high-deductible and a low-deductible insurance policy under different, reasonable accident probabilities
Explain why risk and affordability can matter in a decision alongside the expected value
Prior Knowledge Required
Students should already be comfortable with:
Probabilities of compound events from lists, tables and tree diagrams 7.SP.C.8
Operations with positive and negative rational numbers 7.NS.A.3
Expected value as the mean of a probability distribution HSS.MD.A.2
Building a distribution from theoretical probabilities HSS.MD.A.3
"You can take $50 now, or flip a fair coin: heads you get $120, tails you get nothing. Which do you choose, and why? What would the coin prize have to be for you to change your mind?"
Take a quick vote, then ask students to find the average result of the coin option over many flips: 0.5(120) + 0.5(0) = $60. Many students still choose the sure $50. Keep both answers: the coin option has the higher expected value, but some people prefer a certain amount. That tension is the lesson's theme. Expected value measures the long-run average; the decision also depends on how much risk a person can accept.
Direct Instruction25 minutes
Give the procedure for weighing the outcomes of a decision:
List every outcome of the game or decision, including "win nothing".
Assign a probability to each outcome, from theory or from data, and check that the probabilities add to 1.
Assign a payoff to each outcome: a gain as a positive number and a cost as a negative number. For a game, decide whether you are using prizes (then subtract the cost at the end) or net winnings (prize minus cost).
Compute the expected payoff: E = Σ (payoff × probability).
Compare and decide: compute E for each strategy, pick the higher expected gain or lower expected cost, then ask whether risk or affordability changes the choice.
Official example (a): a scratch-off lottery ticket
An invented $2 scratch-off ticket pays $1,000 with probability 1/5,000, $50 with probability 1/250, $10 with probability 1/50, $4 with probability 1/10 and $2 with probability 1/8. Find the expected net winnings per ticket.
A wheel has 8 equal sections: 1 pays $15, 2 pay $5 and 5 pay nothing. A spin costs $4. Find the expected net winnings and the fair price.
Equation: E(prize) = 15(1/8) + 5(2/8) = $3.125; E(net) = -$0.875; fair price about $3.13; 200 spins earn the booth about $175
Official example (b): high vs low deductible
Low deductible: $1,600 premium, $250 deductible. High deductible: $1,450 premium, $1,000 deductible. A minor accident causes $600 of damage, a major one more than $1,000; assume at most one accident a year. Driver A: P(minor) = 0.10, P(major) = 0.04. Driver B: P(minor) = 0.25, P(major) = 0.10.
A test with 5 choices per question gives +1 for a correct answer, -1/4 for a wrong answer and 0 for a blank. Compare guessing at random with guessing after eliminating one choice.
Equation: Random guess: (1/5)(1) + (4/5)(-1/4) = 0; after eliminating one: (1/4)(1) + (3/4)(-1/4) = 1/16 point, so guess
A business decision under uncertainty
A food cart can set up at an outdoor festival (profit $200 if it rains, $900 if not, P(rain) = 0.3) or at an indoor mall booth (profit $550 for sure).
Equation: E(outdoor) = 0.3(200) + 0.7(900) = $690 > $550, but the outdoor choice risks a $200 day
Use Diagram 1 with Example 3. The two lines show how each policy's expected yearly cost grows with the chance of an accident. For a careful driver such as Driver A, the high deductible is cheaper on average; for a driver with higher accident chances, such as Driver B, the low deductible is. Then ask: "Driver A saves $85 a year on average with the high deductible. What if Driver A has only $300 in savings?" Use Diagram 2 with Example 5 to show how a decision tree organizes the outcomes.
Guided Practice15-20 minutes
Pairs analyze a fast-food game piece (invented odds) that comes free with every meal. They find the expected value of one game piece, then discuss whether the game should affect where they buy lunch.
Prizes on a fast-food game piece (invented odds, guided practice)
Prize
Value
Probability
Free drink
$1.80
1/6
Free fries
$2.40
1/12
Free meal
$8.00
1/100
$100 gift card
$100
1/20,000
No prize
$0
the rest
Answer: E = 1.80(1/6) + 2.40(1/12) + 8(1/100) + 100(1/20,000) = 0.30 + 0.20 + 0.08 + 0.005 = $0.585, so each piece is worth about 59 cents on average. Because the piece is free, this is also the expected net winnings. Ask pairs how the answer would change if the game cost $1 to play. Listen for pairs that add the prize values without weighting them, or that forget to state what "the rest" means as a probability.
Independent Practice15 minutes
Students work alone on three problems:
A club sells 500 raffle tickets at $5 each. Prizes are $600, $200 and two prizes of $50. Find the expected net winnings for one ticket. (E(prize) = 900/500 = $1.80; E(net) = -$3.20.)
A $300 phone has an optional $45 protection plan. Without the plan, there is a 0.12 chance of a $180 repair during the plan period. Compare the expected costs of the two strategies. (Plan: $45; no plan: 0.12 × 180 = $21.60.)
In two or three sentences, describe a situation in which a sensible person would choose the option with the lower expected value, and explain why.
Closure5-10 minutes
Exit ticket: A game costs $1. You roll one die and win $4 if you roll a 6; otherwise you win nothing. (1) Find the expected net winnings. (2) Is the game fair? (3) What price would make it fair? (Answers: 4(1/6) - 1 = -$1/3, about -$0.33; not fair; about $0.67.)
Differentiation Strategies
For Struggling Students
Give a payoff table with columns for outcome, probability, payoff and payoff × probability, so every expected value is the sum of one column
Start with games that are free to play, so the prize and the net winnings are the same, before adding a cost
Use a decision tree for every strategy comparison and have students label each branch with its probability before computing
For Advanced Students
Ask for the accident probability at which the two insurance policies have equal expected cost, and to explain what that value means for choosing a policy
Have students compute the standard deviation of the payoff for two strategies with similar expected values and use it to describe risk
Ask students to change one prize in a game so that the game becomes fair, and to show that the new expected net winnings are 0
Assessment Guidance
What to Look For
Check that every outcome, including "win nothing", has a probability, and that the probabilities add to 1. For games, check that students subtract the cost to play exactly once: either from each prize or from the expected prize, not both. When comparing strategies, look for an expected value for every strategy, a clear statement of which is better for the stated goal (larger gain or smaller cost), and one sentence on risk or affordability. For insurance problems, check that the deductible is capped by the damage: a $600 repair costs $600, not $1,000.
02
Classroom Activities
3 Activities
1
Design a Fundraiser Game
25 minGroups of 3-4
Groups design a booth game for a school fair. The game must look attractive to players but give the booth an expected profit between $0.25 and $1.00 per play. Groups prove this with an expected value calculation and then test the game.
Procedure
Choose a chance device: one or two dice, a spinner drawn on paper, or cards drawn from a bag
Set a price to play and at least two prize levels. List every outcome with its probability and net payoff to the player
Compute the expected net winnings of the player. Adjust prizes or price until the booth's expected profit per play is between $0.25 and $1.00
Play the game 30 times within the group and compare the booth's average profit per play with the expected value
Discussion Questions
Why did your 30 plays not give exactly the expected profit?
If 400 people play during the fair, about how much should the booth earn? What could make the real total very different?
Is a game with a small chance at a big prize more attractive to players than one with a better chance at a small prize, even with the same expected value?
Modification for Distance Learning
Groups build the game in a shared slide deck and use a free online die roller or spinner. Each group posts its payoff table, and another group checks the expected value before playing it online.
2
Which Policy for Which Driver?
20 minPairs
Pairs extend the insurance example to three driver profiles. For each driver, they compute the expected yearly cost of the low-deductible policy ($1,600 premium, $250 deductible) and the high-deductible policy ($1,450 premium, $1,000 deductible). Minor accidents cause $600 of damage and major accidents more than $1,000; assume at most one accident a year.
Driver Profile Cards
Profile 1, long safe record: P(minor) = 0.05, P(major) = 0.02 (low $1,617.50, high $1,500)
Profile 2, average driver: P(minor) = 0.15, P(major) = 0.06 (low $1,652.50, high $1,600)
Profile 3, new driver in heavy traffic: P(minor) = 0.30, P(major) = 0.12 (low $1,705, high $1,750)
Procedure
For each profile, draw a tree with three branches (no accident, minor, major) and the out-of-pocket cost under each policy
Compute both expected yearly costs and circle the lower one
Write a recommendation for each driver that uses the expected values and one other factor, such as savings or peace of mind
Discussion Questions
Why does the high deductible look better for careful drivers?
Where do the accident probabilities come from in real life, and how sure can a driver be about them?
Profile 2 saves about $50 a year with the high deductible. Would you recommend it to someone with no savings? Why or why not?
3
Roll Again? Strategy Showdown
20 minPairs
In this dice game, a player rolls one die and may either keep the number or roll once more and keep the second number, whatever it is. The score is the number kept. Pairs test strategies by playing, then find the expected score of each strategy exactly.
Procedure
Each partner picks a strategy of the form "roll again if the first roll is k or less" and plays 20 rounds, recording the scores
Compare average scores across the class for the strategies "never roll again", "roll again on 1-2", "roll again on 1-3" and "roll again on 1-4"
Compute the expected score of each strategy. Hint: if you roll again, the expected second roll is 3.5
Expected Scores
Never roll again: 3.5
Roll again on 1-2: (4/6)(4.5) + (2/6)(3.5) = 25/6, about 4.17
Roll again on 1-3: (3/6)(5) + (3/6)(3.5) = 4.25, the best strategy
Roll again on 1-4: (2/6)(5.5) + (4/6)(3.5) = 25/6, about 4.17
Challenge Variation
Allow up to two re-rolls. Using the answer for one re-roll (4.25) as the value of rolling again after the first roll, find the best strategy for the first roll and its expected score.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Comparing Two Insurance Policies by Expected Cost
Expected yearly cost (premium plus expected out-of-pocket repairs) for the two policies in Example 3, drawn to scale, when the chance of a major accident is 0.4 times the chance of a minor one. The lines cross where the chance of a minor accident is 3/13, about 0.23. Driver A (0.10) pays less on average with the high deductible; Driver B (0.25) pays less with the low deductible.
Diagram 2: A Decision Tree for Weighing Outcomes
The food cart decision from Example 5. Each chance branch carries a probability and a payoff; the expected value of the outdoor choice, $690, is compared with the certain $550 of the indoor booth.
04
Homework Assignment
~30 min
HSS.MD.B.5 Homework: Expected Payoffs and Strategies
Directions: For each problem, list the outcomes with their probabilities and payoffs, and show the expected value calculation. State whether you are using prizes or net winnings. For comparisons, name the better strategy for the stated goal and write one sentence about risk. All games, prices and probabilities are invented for practice.
Part 1: Expected Payoff for Games of Chance (Problems 1-3)
A sandwich shop gives a free peel-off game piece with every order. A piece wins a free cookie (worth $1.20) with probability 1/5, a free sandwich (worth $7.50) with probability 1/60, or a $500 gift card with probability 1/50,000; otherwise it wins nothing. Find the expected value of one game piece.
A game costs $2. You roll two fair dice. A sum of 12 wins $18, a sum of 7 wins $9, and any other sum wins nothing. (a) Find the expected net winnings. (b) Is the game fair? (c) If the prize for a sum of 7 is lowered to $3, what price would make the game fair?
A $1 lottery ticket pays $5,000 with probability 1/100,000, $100 with probability 1/2,000, $10 with probability 1/100 and $2 with probability 1/10. (a) Find the expected net winnings per ticket. (b) What are the expected net winnings for someone who buys one ticket a week for 52 weeks?
Part 2: Evaluating and Comparing Strategies (Problems 4-6)
Policy L costs $1,320 a year with a $500 deductible. Policy H costs $1,080 a year with a $2,000 deductible. A minor accident causes $1,200 of damage and a major accident causes $9,000; assume at most one accident a year. (a) Compare the expected yearly costs if P(minor) = 0.12 and P(major) = 0.03. (b) Compare them again if P(minor) = 0.30 and P(major) = 0.10. (c) Which driver should choose which policy, and what else should they consider?
A club buys T-shirts for $8 each and sells them for $15. Unsold shirts cannot be returned. Demand will be 80 shirts (probability 0.3), 120 shirts (probability 0.5) or 150 shirts (probability 0.2). Find the expected profit if the club orders 100, 120 or 150 shirts, and recommend an order size.
After a touchdown, a football team can kick for 1 point, which succeeds with probability 0.94, or try a 2-point conversion, which succeeds with probability 0.48. (a) Find the expected points for each choice. (b) Describe a game situation in which the coach should not simply choose the higher expected value.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Outcomes and Probabilities
Every outcome listed, probabilities add to 1
One outcome or probability missing
Outcomes not identified
Payoffs
Gains and costs signed correctly; cost to play subtracted once
Cost ignored or subtracted twice
Payoffs missing or wrong
Expected Value
Weighted sum shown and correct for every strategy
Method correct with an arithmetic error
No expected value or unweighted average
Decision and Reasoning
Better strategy named for the goal, with a sentence on risk
Choice stated without reasoning about risk
No decision
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. All games, prices and probabilities are invented for practice.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A spinner has 4 equal sections. One section wins $3 and the others win nothing. It costs $1 to spin. What are the expected net winnings per spin?
Answer: A
E(prize) = 3(1/4) = $0.75, and subtracting the $1 cost gives -$0.25 per spin. Choice B forgets the cost to play. Choice D subtracts the cost from the top prize without weighting it by its probability.
Question 2 of 20 · Multiple Choice
What does it mean to weigh the outcomes of a decision by finding an expected value?
Answer: C
The expected value weights each payoff by how likely it is and adds the results. Choice A ignores probability, and choice B ignores the size of the payoffs. Choice D is the unweighted average, which treats every outcome as equally likely.
Question 3 of 20 · Multiple Choice
A club sells 1,000 raffle tickets for $2 each. There is one $800 prize and four $50 prizes. What are the expected net winnings for one ticket?
Answer: B
E(prize) = (800 + 4 × 50)/1,000 = $1.00, so E(net) = 1.00 - 2 = -$1.00. Choice A is the expected prize before subtracting the ticket price. Choice C leaves out the four $50 prizes. Choice D is the total prize money.
Question 4 of 20 · Multiple Choice
Which of these games is fair?
Answer: D
A game is fair when the expected net winnings are 0. In D, 3(1/3) - 1 = 0. The others are unfavorable to the player: A gives 5(1/4) - 2 = -$0.75, B gives 2(1/3) - 1 = -$1/3 and C gives 10(1/4) - 3 = -$0.50. Choosing C because its prize is largest ignores the probability.
Question 5 of 20 · Multiple Choice
A game pays $12 with probability 1/8 and nothing otherwise. What price to play would make the game fair?
Answer: C
The fair price equals the expected prize: 12(1/8) = $1.50. Choice A is the prize itself, which would make the game unfavorable to the player. Choice B confuses the 8 in the probability with a dollar amount.
Question 6 of 20 · Multiple Choice
A free restaurant game piece wins a drink worth $2 with probability 1/4 or a burger worth $6 with probability 1/20; otherwise it wins nothing. What is the expected value of one game piece?
Answer: A
E = 2(1/4) + 6(1/20) = 0.50 + 0.30 = $0.80. Choice B adds the prize values without probabilities. Choice C adds the two probabilities. Choice D averages the two prizes.
Question 7 of 20 · Multiple Choice
The expected prize from a $1 lottery ticket is $0.55. What are the expected net winnings from buying 100 tickets?
Answer: D
Each ticket has E(net) = 0.55 - 1 = -$0.45, so 100 tickets have E(net) = 100(-0.45) = -$45. Choice A is the result for one ticket. Choice B is the expected total prize money before paying for the tickets. Choice C treats the $55 of expected prizes as a loss and forgets the $100 paid for the tickets.
Question 8 of 20 · Multiple Choice
Policy A costs $900 a year with a $500 deductible. Policy B costs $1,050 a year with a $100 deductible. A driver has a 0.2 chance of one accident a year, with damage above $500. Which policy has the lower expected yearly cost?
Answer: B
Policy A: 900 + 0.2(500) = $1,000. Policy B: 1,050 + 0.2(100) = $1,070. Choice A looks only at the deductible, and choice C compares only the premiums.
Question 9 of 20 · Multiple Choice
A test gives +4 points for a correct answer, -1 for a wrong answer and 0 for a blank. A student eliminates 2 of the 5 choices and guesses at random among the other 3. What is the expected score for the guess?
Answer: C
E = 4(1/3) + (-1)(2/3) = 4/3 - 2/3 = 2/3 point, which is better than 0 for a blank, so guessing is the better strategy. Choice A is the value of a random guess among all 5 choices. Choice B forgets the penalty for a wrong answer.
Question 10 of 20 · Multiple Choice
Location A gives a profit of $400 with probability 0.6 and $100 with probability 0.4. Location B gives $300 for sure. Which has the higher expected profit?
Answer: A
E(A) = 0.6(400) + 0.4(100) = 240 + 40 = $280, less than $300. Choice B looks only at the best case. Choice C averages 400 and 100 without the probabilities and also names the wrong location.
Question 11 of 20 · Multiple Choice
Why might a person buy insurance even though the expected cost of insurance is higher than the expected cost of going without it?
Answer: D
Insurers set premiums above expected claims, so for most customers the expected cost with insurance is higher. People buy it to avoid a large, unaffordable loss. Choice A is false: if it were true, the insurer would lose money on average.
Question 12 of 20 · Multiple Choice
A game costs $4. You roll one die and win the number of dollars shown. What are the expected net winnings?
Answer: B
The expected roll is (1 + 2 + 3 + 4 + 5 + 6)/6 = $3.50, so E(net) = 3.50 - 4 = -$0.50. Choice A forgets the cost. Choice C subtracts in the wrong order.
Question 13 of 20 · Multiple Choice
A basketball team trails by 2 with seconds left. A 2-point shot goes in with probability 0.50, and then the team wins in overtime with probability 0.50. A 3-point shot goes in with probability 0.35 and wins the game. Which strategy gives the higher chance of winning?
Answer: C
With a payoff of 1 for a win and 0 for a loss, the expected payoff is the probability of winning. The 2-point strategy wins with probability 0.50 × 0.50 = 0.25, and the 3-point strategy with 0.35. Choice A forgets that the team must still win overtime. Choice D adds probabilities that should be multiplied.
Question 14 of 20 · Multiple Choice
Policy L costs $1,200 a year with a $250 deductible. Policy H costs $1,000 a year with a $1,000 deductible. Every accident causes more than $1,000 of damage, and there is at most one a year. For what accident probability p do the policies have the same expected cost?
Answer: A
Set 1,200 + 250p = 1,000 + 1,000p, so 200 = 750p and p = 4/15, about 0.27. For a smaller p, Policy H is cheaper on average. Choice B divides 200 by 1,000 and ignores the $250 deductible.
Question 15 of 20 · Short Answer
A booth charges $1 to roll two dice. If the dice show doubles, the player wins $5; otherwise the player wins nothing. Find the expected net winnings per play and the booth's expected profit from 300 plays.
P(doubles) = 6/36 = 1/6. E(net) = 5(1/6) - 1 = -1/6, about -$0.17 per play. The booth expects to gain 1/6 of a dollar per play, so about 300 × 1/6 = $50 from 300 plays.
Question 16 of 20 · Short Answer
An invented $3 scratch-off ticket pays $3,000 with probability 1/20,000, $60 with probability 1/400, $15 with probability 1/50 and $6 with probability 1/10. Find the expected net winnings per ticket.
E(prize) = 3,000/20,000 + 60/400 + 15/50 + 6/10 = 0.15 + 0.15 + 0.30 + 0.60 = $1.20. E(net) = 1.20 - 3 = -$1.80 per ticket. On average a buyer loses $1.80 of every $3 spent.
Question 17 of 20 · Short Answer
Job A pays a $3,000 salary for the summer. Job B pays only commission: $1,500 with probability 0.3, $3,200 with probability 0.5 and $5,000 with probability 0.2. Compare the jobs using expected value and risk.
E(B) = 0.3(1,500) + 0.5(3,200) + 0.2(5,000) = 450 + 1,600 + 1,000 = $3,050, only $50 more than Job A. Job B has a 0.3 chance of earning half as much. A student who needs a set amount, for example for tuition, may reasonably prefer the certain $3,000 of Job A; a student who can accept the risk may choose Job B.
Question 18 of 20 · Short Answer
Give two reasons why the option with the higher expected value is not always the best decision for a person.
Sample answer: (1) Risk: the expected value is a long-run average, but a person may face the decision only once, and the worst outcome may be unacceptable, such as a large loss they cannot pay. (2) Other values: the payoffs may leave out things that matter, such as time, stress, safety or enjoyment. A third reason is that the probabilities may be estimates that are not reliable.
Question 19 of 20 · Short Answer
A spinner has 5 equal sections, and one section wins a prize. The game costs $2 to play. What prize would make the game fair? Show your reasoning.
For a fair game, the expected prize equals the cost: prize × (1/5) = 2, so the prize must be $10. Check: E(net) = 10(1/5) - 2 = 0.
Question 20 of 20 · Short Answer
Policy L costs $1,100 a year with a $300 deductible. Policy H costs $850 a year with a $1,500 deductible. A minor accident causes $900 of damage (probability 0.15) and a major accident causes $6,000 (probability 0.05); assume at most one accident a year. Compare the expected yearly costs and give a recommendation.
L: 1,100 + 0.20(300) = $1,160. H: 850 + 0.15(900) + 0.05(1,500) = 850 + 135 + 75 = $1,060, since a $900 repair is below the deductible. H is $100 cheaper on average. Recommend H for a driver who could pay a $1,500 bill; a driver without that much in savings may prefer L.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.MD.B.5 mean?
HSS.MD.B.5 means students use expected value to make decisions: they assign a probability to each payoff of a choice, find the expected payoff and compare options. Part a applies this to games of chance, and part b to comparing strategies. It is marked (+), so it is part of advanced courses.
What is the difference between HSS.MD.B.5a and HSS.MD.B.5b?
Part a finds one expected payoff, and part b compares several. In part a, students compute the expected payoff of a single game, such as a lottery ticket or a restaurant game piece. In part b, they compute an expected value for each strategy, such as two insurance policies or two ways to answer a test question, and use the results to choose.
What is an expected payoff?
It is the long-run average amount gained or lost per play or per decision. To find it, multiply each possible payoff by its probability and add. A negative expected payoff means the player loses money on average over many plays.
What makes a game of chance fair?
A game is fair when the expected net winnings of the player are 0. Equivalently, the price to play equals the expected prize. Most real games of chance, including lotteries and casino games, are designed to have negative expected net winnings for the player.
Why do people buy lottery tickets if the expected value is negative?
Because they value things the expected value leaves out. A ticket costs little, the chance of a large prize is exciting, and some people see it as entertainment. The expected value still tells students what happens to a regular buyer over time: on average, they get back less than they spend.
How do you compare a high-deductible and a low-deductible insurance policy?
Find each policy's expected yearly cost: the premium plus the expected out-of-pocket cost of accidents. For each type of accident, the out-of-pocket cost is the smaller of the damage and the deductible, times the accident's probability. Then repeat with different reasonable accident probabilities, as the standard suggests, to see when the better choice changes.
Should students always choose the option with the higher expected value?
No: expected value is one input to the decision. A person who faces a decision only once may care about the worst case, and some outcomes, such as a bill they cannot pay, matter more than their dollar amount suggests. Good answers state the expected values and then explain the choice in context.
Should students use the prizes or the net winnings when a game costs money?
Either works, as long as the cost is counted exactly once. Students can subtract the cost from each prize and use the net payoffs, or find the expected prize and subtract the cost at the end. A common error is to do both, which subtracts the cost twice.
What course teaches HSS.MD.B.5?
It is usually taught in a statistics course or in Precalculus, and sometimes in an advanced Algebra II course. As a (+) standard, it goes beyond the core that all students take, and it connects closely to personal finance topics such as insurance and loans.
What mistakes do students make with expected payoff?
Common mistakes are forgetting the cost to play, leaving out the "win nothing" outcome, averaging the payoffs without probabilities and using a deductible larger than the damage. In comparisons, some students compute only one strategy's expected value or compare premiums alone.
07
Related Standards
5 standards
These standards connect to HSS.MD.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.SP.C.8Prerequisite
Find probabilities of compound events using lists, tables, tree diagrams and simulation
