Understanding Expected Value Concepts Quiz

1) What does the expected value of a random variable represent?

The long-run average outcome if the experiment is repeated many times

The maximum possible outcome

The outcome that occurs most frequently

The minimum possible outcome

Expected value is the long-run average result you’d get if you repeated the experiment many times.

Explanation

Expected value is the long-run average result you’d get if you repeated the experiment many times.

2) A random variable X can take values 1, 2, or 3, each with equal probability. What is the expected value of X?

1

2

3

6

Each value has probability 1/3. E(X) = (1 + 2 + 3) / 3 = 6 / 3 = 2.

Explanation

Each value has probability 1/3. E(X) = (1 + 2 + 3) / 3 = 6 / 3 = 2.

3) Which of the following best describes a random variable?

A variable that is always equal to its expected value

A numerical value assigned to each outcome in a sample space

A variable that changes randomly without any pattern

A number that represents the most likely outcome

A random variable assigns a number (like 0, 1, 2, …) to each outcome in the sample space.

Explanation

A random variable assigns a number (like 0, 1, 2, …) to each outcome in the sample space.

4) If you flip a fair coin and win $5 for heads and lose $3 for tails, what is the expected value of your winnings?

$5

$1

$2

$1

P(H) = 0.5, P(T) = 0.5. E = 0.5(5) + 0.5(−3) = 2.5 − 1.5 = 1.

Explanation

P(H) = 0.5, P(T) = 0.5. E = 0.5(5) + 0.5(−3) = 2.5 − 1.5 = 1.

5) The expected value of a random variable is always:

A whole number

One of the possible outcomes

Greater than zero

A weighted average of all possible outcomes

Expected value is found by multiplying each outcome by its probability and adding them — a weighted average.

Explanation

Expected value is found by multiplying each outcome by its probability and adding them — a weighted average.

6) What is the expected value of the prize money from one spin?

$1.85

$2.10

$1.75

$2.50

E = 0(0.50) + 2(0.30) + 5(0.15) + 10(0.05) = 0 + 0.6 + 0.75 + 0.5 = 1.85.

Explanation

E = 0(0.50) + 2(0.30) + 5(0.15) + 10(0.05) = 0 + 0.6 + 0.75 + 0.5 = 1.85.

7) If the game costs $3 to play, what is the expected profit (or loss) per game?

Profit of $1.15

Loss of $1.15

Loss of $0.90

Profit of $0.90

Expected prize = $1.85. Expected profit = 1.85 − 3.00 = −1.15 → loss of $1.15.

Explanation

Expected prize = $1.85. Expected profit = 1.85 − 3.00 = −1.15 → loss of $1.15.

8) Which prize amount has the greatest contribution to the expected value?

$0

$2

$10

$5

Contribution = prize × probability. $2 → 0.60, $5 → 0.75, $10 → 0.50. Largest is 0.75, from $5.

Explanation

Contribution = prize × probability. $2 → 0.60, $5 → 0.75, $10 → 0.50. Largest is 0.75, from $5.

9) If 1,000 people play this game, approximately how much total prize money would you expect to be paid out?

$1,000

$1,850

$2,100

$5,000

Expected prize per game = $1.85. For 1,000 games: 1.85 × 1000 = $1,850.

Explanation

Expected prize per game = $1.85. For 1,000 games: 1.85 × 1000 = $1,850.

10) If the carnival wants to make an average profit of $1 per game, how much should they charge to play?

$2.85

$3.10

$2.10

$4.00

Let price = P. Profit per game = P − 1.85. Set P − 1.85 = 1 → P = 2.85.

Explanation

Let price = P. Profit per game = P − 1.85. Set P − 1.85 = 1 → P = 2.85.

11) A probability distribution shows that a random variable Y has values 10, 20, and 30 with probabilities 0.2, 0.5, and 0.3 respectively. What is E(Y)?

20

23

25

21

E(Y) = 10(0.2) + 20(0.5) + 30(0.3) = 2 + 10 + 9 = 21.

Explanation

E(Y) = 10(0.2) + 20(0.5) + 30(0.3) = 2 + 10 + 9 = 21.

12) Which statement about expected value is TRUE?

The expected value must be one of the possible outcomes

The expected value can be a number that is not a possible outcome

The expected value is always the median of the distribution

The expected value is always positive

Expected value is an average; it can be between outcomes (like 3.5 on a die), not necessarily one of them.

Explanation

Expected value is an average; it can be between outcomes (like 3.5 on a die), not necessarily one of them.

13) A random variable has an expected value of 7.5. This means:

The outcome will always be 7.5

The most common outcome is 7.5

Over many trials, the average outcome will be close to 7.5

The outcome will be between 7 and 8

Expected value describes the average over many repetitions, not a guarantee for a single trial.

Explanation

Expected value describes the average over many repetitions, not a guarantee for a single trial.

14) In a probability distribution, if all outcomes have equal probability, the expected value equals:

The arithmetic mean of all outcomes

The largest outcome

The smallest outcome

Zero

With equal probabilities, each outcome is weighted equally, so E = regular (arithmetic) mean of the values.

Explanation

With equal probabilities, each outcome is weighted equally, so E = regular (arithmetic) mean of the values.

15) What is the expected number of pets per student?

1.0

1.3

2.0

E(X) = 0(0.25) + 1(0.40) + 2(0.20) + 3(0.10) + 4(0.05) = 0 + 0.40 + 0.40 + 0.30 + 0.20 = 1.30.

Explanation

E(X) = 0(0.25) + 1(0.40) + 2(0.20) + 3(0.10) + 4(0.05) = 0 + 0.40 + 0.40 + 0.30 + 0.20 = 1.30.

16) What is the most likely number of pets a randomly selected student owns?

1

2

1.3

0

The highest probability is 0.40 at X = 1, so 1 pet is the most likely (the mode).

Explanation

The highest probability is 0.40 at X = 1, so 1 pet is the most likely (the mode).

17) How does the expected value (1.3 pets) differ from the mode (1 pet)?

They represent the same concept

The expected value is always larger than the mode

The mode is more accurate than the expected value

The mode is the most frequent outcome; the expected value is the weighted average

Mode = most frequent value. Expected value = average using probabilities as weights.

Explanation

Mode = most frequent value. Expected value = average using probabilities as weights.

18) If there are 200 students in the school, approximately how many total pets would you expect all students to own?

200

130

260

300

Expected pets per student = 1.3. Total expected pets = 200 × 1.3 = 260.

Explanation

Expected pets per student = 1.3. Total expected pets = 200 × 1.3 = 260.

19) A fair six-sided die is rolled. What is the expected value of the roll?

3.0

3.5

4.0

6.0

Outcomes 1–6 each with probability 1/6: E = (1+2+3+4+5+6)/6 = 21/6 = 3.5.

Explanation

Outcomes 1–6 each with probability 1/6: E = (1+2+3+4+5+6)/6 = 21/6 = 3.5.

20) Which real-world scenario is BEST modeled using expected value?

Predicting the exact outcome of a single coin flip

Determining the winner of a single game

Calculating the average insurance payout an insurance company expects per policy

Finding the maximum possible profit from a business venture

Expected value is used to find average outcomes over many cases, like average insurance payout per policy.

Explanation

Expected value is used to find average outcomes over many cases, like average insurance payout per policy.