Expected Value

Show your work for each problem.

1.
What does the expected value of a random experiment represent?
1. a)The long-run average outcome
2. b)The minimum possible outcome
3. c)The maximum possible outcome
4. d)The most likely outcome
Answer: The long-run average outcome
The expected value is the long-run average outcome. If you repeat the experiment many times, your average result will approach the expected value.
2.
What is the expected value when flipping a fair coin where heads = 1 and tails = 0?
Answer: 0.5
. The expected value is 0.5, which is halfway between 0 and 1.
3.
A fair six-sided die is rolled. What is the expected value?
1. a)3
2. b)3.5
3. c)4
4. d)3.25
Answer: 3.5
. Even though you can never roll 3.5, this is the average you expect over many rolls.
4.
A spinner has 4 equal sections numbered 2, 4, 6, and 8. Find the expected value.
Answer: 5
1. What is the probability of landing on each section? 1/4
2. What is the sum of all outcomes? (2+4+6+8) 20
3. Calculate the expected value: sum divided by number of outcomes 5
5.
A biased coin has a 60% chance of heads (win 3 euros) and 40% chance of tails (lose 2 euros). What is the expected value?
Answer: 1
euro. On average, you win 1 euro per flip.
6.
A game costs 5 euros to play. You roll a die and win 12 euros if you roll a 6, otherwise you win nothing. What is the expected profit?
1. a)0 euros
2. b)-3 euros
3. c)-2 euros
4. d)2 euros
Answer: -3 euros
euros. euros. You lose 3 euros on average.
7.
A lottery ticket costs 2 euros. There's a 5% chance of winning 20 euros and a 95% chance of winning nothing. Calculate the expected profit.
Answer: -1
1. Calculate (expected winnings from prize) 1
2. Calculate (expected winnings from no prize) 0
3. What is the total expected winnings? 1
4. Subtract the ticket cost (2 euros) to get expected profit -1
8.
If a game has an expected value of -2 euros, what does this mean?
1. a)You will always lose 2 euros
2. b)You can never win
3. c)On average, you lose 2 euros per game
4. d)The game costs 2 euros to play
Answer: On average, you lose 2 euros per game
A negative expected value means you lose money on average. You might win sometimes and lose sometimes, but over many games, you'll lose about 2 euros per game on average.
9.
A spinner has 3 sections: red (50%), blue (30%), yellow (20%). Red wins 4 euros, blue wins 2 euros, yellow loses 5 euros. What is the expected value?
Answer: 1.6
euros. This is a favorable game with positive expected value.
10.
An insurance company charges 200 euros per year. 3% of customers file claims averaging 4000 euros. What is the expected profit per customer?
Answer: 80
1. Calculate expected payout: 120
2. What is the annual premium collected? 200
3. Calculate expected profit: premium minus expected payout 80
11.
In European roulette, betting 10 euros on a single number pays 350 euros if you win (35:1 odds). There are 37 numbers. What is the expected value of this bet?
Answer: -0.27
euros. The house edge is about 2.7%.
12.
A game show offers: keep 1000 euros OR flip a coin where heads wins 2500 euros and tails wins 0 euros. Which has higher expected value and by how much?
Answer: 250
1. What is the expected value of keeping 1000 euros? 1000
2. Calculate expected value of the coin flip: 1250
3. Which option has higher expected value? flip
4. By how much is it higher? 250
13.
Investment A: 40% chance of +30%, 60% chance of -10%. Investment B: 80% chance of +10%, 20% chance of +5%. Which has higher expected return?
1. a)Investment A (9%)
2. b)They are equal
3. c)Investment A (6%)
4. d)Investment B (9%)
Answer: Investment B (9%)
A: . B: . B has higher expected return (9% vs 6%).
14.
A company manufactures products. 2% are defective and cost 500 euros each to replace. Good products earn 20 euros profit. What is the expected profit per product?
Answer: 9.6
1. What is the probability of a good product? 0.98
2. Calculate expected profit from good products: 19.6
3. Calculate expected loss from defects: -10
4. Add to find total expected profit 9.6
15.
Why can't you expect to roll 3.5 on a single die roll, even though the expected value is 3.5?
1. a)The expected value is wrong
2. b)3.5 is not a possible outcome on a die
3. c)You need to roll twice
4. d)The die is unfair
Answer: 3.5 is not a possible outcome on a die
A die only has faces numbered 1, 2, 3, 4, 5, 6. You can never roll 3.5! The expected value is a theoretical average that emerges over many rolls, not a single outcome.
16.
Two dice are rolled and their sum is recorded. What is the expected value of the sum?
Answer: 7
. This is why 7 is the most common sum when rolling two dice!