Calculating Expected Value For Discrete Distributions Quiz

1) A random variable Y has E(Y)=8. If 5 is added to each value, what is the expected value of the new distribution?

8

10

13

40

8 + 5 = 13.

Explanation

8 + 5 = 13.

2) A student calculates E(X)=7.8 for a discrete random variable. Which statement is most accurate?

Expected value must be a whole number

The expected value is valid and represents the mean of the distribution

The expected value must equal one of the possible XXX values

The calculation is only correct if all probabilities are equal

Expected value can be any real number and represents the mean.

Explanation

Expected value can be any real number and represents the mean.

3) For random variable Z: P(Z=−2)=0.3, P(Z=0)=0.4, P(Z=4)=0.3. What is E(Z)?

0

0.6

1.2

2

E(Z)=0.6.

Explanation

E(Z)=0.6.

4) After 200 spins, approximately how many total points should a player expect to accumulate?

500

580

700

600

3 × 200 = 600 points.

Explanation

3 × 200 = 600 points.

5) What is the expected value of points per spin?

3.0

3.1

3.5

4.0

E(X) = 3.0.

Explanation

E(X) = 3.0.

6) Which statement about expected value is TRUE?

The expected value represents a long-run average

The expected value must be one of the possible outcomes

The expected value is always equal to the median

The expected value is the most likely outcome

Expected value is the long-run average result over many trials.

Explanation

Expected value is the long-run average result over many trials.

7) A game costs $3 to play. P($10)=0.2, P($0)=0.8. What is the expected net gain/loss per game?

Gain of $2

Break even

Loss of $3

Loss of $1

Expected prize = 2; net = 2 − 3 = −1.

Explanation

Expected prize = 2; net = 2 − 3 = −1.

8) If the company’s goal is E[defects] < 0.30 per item, is this goal currently being met?

No, because 0.42 > 0.30

Yes, because 0.42 < 0.30

Yes, because most items have 0 defects

Cannot be determined

Expected defects = 0.42 > 0.30.

Explanation

Expected defects = 0.42 > 0.30.

9) In a batch of 500 items, approximately how many total defects should the inspector expect to find?

175

250

210

300

0.42 × 500 = 210.

Explanation

0.42 × 500 = 210.

10) What is the expected number of defects per item?

0.35

0.42

0.50

1.00

E(X)=0.42.

Explanation

E(X)=0.42.

11) What is the expected value (mean) of tickets won per game?

10 tickets

12 tickets

15 tickets

8 tickets

E(X) = 0·0.35 + 5·0.30 + 10·0.20 + 20·0.10 + 50·0.05 = 8 tickets.

Explanation

E(X) = 0·0.35 + 5·0.30 + 10·0.20 + 20·0.10 + 50·0.05 = 8 tickets.

12) A distribution has E(X)=12. If each value of X is multiplied by 3, what is the new expected value?

36

12

15

48

3 × 12 = 36.

Explanation

3 × 12 = 36.

13) If someone buys 10 tickets, what is their expected net gain or loss?

Loss of $5

Loss of $15

Loss of $25

Gain of $10

Net loss per ticket = $1.50 → 10 × 1.50 = $15 loss.

Explanation

Net loss per ticket = $1.50 → 10 × 1.50 = $15 loss.

14) What is the expected net gain (or loss) for a person who buys one ticket?

Gain of $1.50

Break even

Loss of $1.50

Loss of $2.50

Expected prize = 3.50, cost = 5 → net = −1.50.

Explanation

Expected prize = 3.50, cost = 5 → net = −1.50.

15) What is the expected value of a single raffle ticket (prize winnings only)?

$2.50

$3.00

$3.50

$4.00

Expected prize = $3.50.

Explanation

Expected prize = $3.50.

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

3

4

3.25

3.5

E = (1+2+3+4+5+6)/6 = 3.5.

Explanation

E = (1+2+3+4+5+6)/6 = 3.5.

17) A random variable X has: P(X=2)=0.4, P(X=5)=0.3, P(X=8)=0.3. What is E[X]?

5

4.7

4.5

6

E(X) = 2·0.4 + 5·0.3 + 8·0.3 = 4.7.

Explanation

E(X) = 2·0.4 + 5·0.3 + 8·0.3 = 4.7.

18) What is the probability of winning more tickets than the expected value in a single game?

0.10

0.15

0.35

0.50

More than 8 tickets means 10, 20, or 50. P = 0.20 + 0.10 + 0.05 = 0.35.

Explanation

More than 8 tickets means 10, 20, or 50. P = 0.20 + 0.10 + 0.05 = 0.35.

19) Which interpretation best describes the expected value in this context?

The minimum number of tickets a player will win

The average number of tickets won per game over many plays

The most common number of tickets won in a single game

The median number of tickets won per game

Expected value is the long-run average number of tickets per game over many plays.

Explanation

Expected value is the long-run average number of tickets per game over many plays.

20) If a player plays this game 100 times, approximately how many total tickets should they expect to win?

800 tickets

750 tickets

850 tickets

1,000 tickets

Expected tickets per game = 8. For 100 games: 8 × 100 = 800 tickets.

Explanation

Expected tickets per game = 8. For 100 games: 8 × 100 = 800 tickets.

Calculating Expected Value for Discrete Distributions Quiz

1) A random variable Y has E(Y)=8. If 5 is added to each value, what is the expected value of the new distribution?

2)

What first name or nickname would you like us to use?

2) A student calculates E(X)=7.8 for a discrete random variable. Which statement is most accurate?

3) For random variable Z: P(Z=−2)=0.3, P(Z=0)=0.4, P(Z=4)=0.3. What is E(Z)?

4) After 200 spins, approximately how many total points should a player expect to accumulate?

5) What is the expected value of points per spin?

6) Which statement about expected value is TRUE?

7) A game costs $3 to play. P($10)=0.2, P($0)=0.8. What is the expected net gain/loss per game?

8) If the company’s goal is E[defects] < 0.30 per item, is this goal currently being met?

9) In a batch of 500 items, approximately how many total defects should the inspector expect to find?

10) What is the expected number of defects per item?

11) What is the expected value (mean) of tickets won per game?

12) A distribution has E(X)=12. If each value of X is multiplied by 3, what is the new expected value?

13) If someone buys 10 tickets, what is their expected net gain or loss?

14) What is the expected net gain (or loss) for a person who buys one ticket?

15) What is the expected value of a single raffle ticket (prize winnings only)?

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

17) A random variable X has: P(X=2)=0.4, P(X=5)=0.3, P(X=8)=0.3. What is E[X]?

18) What is the probability of winning more tickets than the expected value in a single game?

19) Which interpretation best describes the expected value in this context?

20) If a player plays this game 100 times, approximately how many total tickets should they expect to win?