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Expected Value worksheet

Grades 11–12 · CCSS HSS.MD.A.2, HSS.MD.B.5

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Outcome values (x): -6, 0, 5, 45
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
2.
Problem 2.
- Outcome values (x): -10, 0, 5, 50
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
3.
Problem 3.
- Outcome values (x): -10, 0, 6, 40
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
4.
Problem 4.
- Outcome values (x): -11, 3, 6, 50
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
5.
Problem 5.
- Outcome values (x): -8, 0, 6, 45
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
6.
Problem 6.
- Outcome values (x): -7, 0, 4, 58
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
7.
Problem 7.
- Outcome values (x): -12, 0, 4, 48
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
8.
Problem 8.
- Outcome values (x): -9, 0, 6, 75
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
9.
Problem 9.
- Outcome values (x): -10, 1, 6, 46
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer
10.
Problem 10.
- Outcome values (x): -9, 2, 5, 52
- Probabilities (P), in the same order: 0.4, 0.3, 0.2, 0.1
Find: Expected value, E[X]
Answer

Answer key: Expected Value worksheet

1. Expected value, E[X]: 3.1
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −6 \times 0.4 + 0 \times 0.3 + 5 \times 0.2 + 45 \times 0.1$ $= 3.1$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−6 - 3.1)}^{2} + 0.3 \times {(0 - 3.1)}^{2} + 0.2 \times {(5 - 3.1)}^{2} + 0.1 \times {(45 - 3.1)}^{2}$ $= 212.29$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{212.29}$ $= 14.5701750161$
2. Expected value, E[X]: 2
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −10 \times 0.4 + 0 \times 0.3 + 5 \times 0.2 + 50 \times 0.1$ $= 2$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−10 - 2)}^{2} + 0.3 \times {(0 - 2)}^{2} + 0.2 \times {(5 - 2)}^{2} + 0.1 \times {(50 - 2)}^{2}$ $= 291$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{291}$ $= 17.0587221092$
3. Expected value, E[X]: 1.2
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −10 \times 0.4 + 0 \times 0.3 + 6 \times 0.2 + 40 \times 0.1$ $= 1.2$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−10 - 1.2)}^{2} + 0.3 \times {(0 - 1.2)}^{2} + 0.2 \times {(6 - 1.2)}^{2} + 0.1 \times {(40 - 1.2)}^{2}$ $= 205.76$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{205.76}$ $= 14.3443368616$
4. Expected value, E[X]: 2.7
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −11 \times 0.4 + 3 \times 0.3 + 6 \times 0.2 + 50 \times 0.1$ $= 2.7$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−11 - 2.7)}^{2} + 0.3 \times {(3 - 2.7)}^{2} + 0.2 \times {(6 - 2.7)}^{2} + 0.1 \times {(50 - 2.7)}^{2}$ $= 301.01$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{301.01}$ $= 17.3496397657$
5. Expected value, E[X]: 2.5
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −8 \times 0.4 + 0 \times 0.3 + 6 \times 0.2 + 45 \times 0.1$ $= 2.5$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−8 - 2.5)}^{2} + 0.3 \times {(0 - 2.5)}^{2} + 0.2 \times {(6 - 2.5)}^{2} + 0.1 \times {(45 - 2.5)}^{2}$ $= 229.05$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{229.05}$ $= 15.1343979068$
6. Expected value, E[X]: 3.8
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −7 \times 0.4 + 0 \times 0.3 + 4 \times 0.2 + 58 \times 0.1$ $= 3.8$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−7 - 3.8)}^{2} + 0.3 \times {(0 - 3.8)}^{2} + 0.2 \times {(4 - 3.8)}^{2} + 0.1 \times {(58 - 3.8)}^{2}$ $= 344.76$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{344.76}$ $= 18.5677139142$
7. Expected value, E[X]: 0.8
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −12 \times 0.4 + 0 \times 0.3 + 4 \times 0.2 + 48 \times 0.1$ $= 0.8$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−12 - 0.8)}^{2} + 0.3 \times {(0 - 0.8)}^{2} + 0.2 \times {(4 - 0.8)}^{2} + 0.1 \times {(48 - 0.8)}^{2}$ $= 290.56$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{290.56}$ $= 17.0458206021$
8. Expected value, E[X]: 5.1
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −9 \times 0.4 + 0 \times 0.3 + 6 \times 0.2 + 75 \times 0.1$ $= 5.1$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−9 - 5.1)}^{2} + 0.3 \times {(0 - 5.1)}^{2} + 0.2 \times {(6 - 5.1)}^{2} + 0.1 \times {(75 - 5.1)}^{2}$ $= 576.09$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{576.09}$ $= 24.0018749268$
9. Expected value, E[X]: 2.1
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −10 \times 0.4 + 1 \times 0.3 + 6 \times 0.2 + 46 \times 0.1$ $= 2.1$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−10 - 2.1)}^{2} + 0.3 \times {(1 - 2.1)}^{2} + 0.2 \times {(6 - 2.1)}^{2} + 0.1 \times {(46 - 2.1)}^{2}$ $= 254.69$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{254.69}$ $= 15.9590099944$
10. Expected value, E[X]: 3.2
1. Check that the 4 probabilities add up to 1
  $0.4 + 0.3 + 0.2 + 0.1$ $= 1$
2. Multiply each outcome by its probability and add the products to get the expected value
  $E[X]$ $= −9 \times 0.4 + 2 \times 0.3 + 5 \times 0.2 + 52 \times 0.1$ $= 3.2$
3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them
  $Var(X)$ $= 0.4 \times {(−9 - 3.2)}^{2} + 0.3 \times {(2 - 3.2)}^{2} + 0.2 \times {(5 - 3.2)}^{2} + 0.1 \times {(52 - 3.2)}^{2}$ $= 298.76$
4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units
  $σ$ $= \sqrt{298.76}$ $= 17.2846752934$