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Pythagorean Expectation worksheet

Grade 6 · CCSS 6.EE.A.1, 6.EE.A.2.c

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

1.
Problem 1.
- Runs (or points) scored: 750
- Runs (or points) allowed: 675
- Exponent: 2 (original)
- Games played: 190
- Actual wins (optional): 70
Find: Expected winning percentage
Answer
2.
Problem 2.
- Runs (or points) scored: 740
- Runs (or points) allowed: 550
- Exponent: 1.83 (refined baseball)
- Games played: 140
- Actual wins (optional): 100
Find: Expected winning percentage
Answer
3.
Problem 3.
- Runs (or points) scored: 900
- Runs (or points) allowed: 540
- Exponent: Custom
- Custom exponent: 2
- Games played: 160
- Actual wins (optional): 110
Find: Expected winning percentage
Answer
4.
Problem 4.
- Runs (or points) scored: 1,100
- Runs (or points) allowed: 750
- Exponent: Custom
- Custom exponent: 3
- Games played: 180
- Actual wins (optional): 110
Find: Expected winning percentage
Answer
5.
Problem 5.
- Runs (or points) scored: 600
- Runs (or points) allowed: 550
- Exponent: 2 (original)
- Games played: 150
- Actual wins (optional): 80
Find: Expected winning percentage
Answer
6.
Problem 6.
- Runs (or points) scored: 520
- Runs (or points) allowed: 680
- Exponent: 2 (original)
- Games played: 150
- Actual wins (optional): 120
Find: Expected winning percentage
Answer
7.
Problem 7.
- Runs (or points) scored: 675
- Runs (or points) allowed: 700
- Exponent: 1.83 (refined baseball)
- Games played: 220
- Actual wins (optional): 65
Find: Expected winning percentage
Answer
8.
Problem 8.
- Runs (or points) scored: 900
- Runs (or points) allowed: 650
- Exponent: 1.83 (refined baseball)
- Games played: 120
- Actual wins (optional): 88
Find: Expected winning percentage
Answer
9.
Problem 9.
- Runs (or points) scored: 720
- Runs (or points) allowed: 775
- Exponent: 1.83 (refined baseball)
- Games played: 160
- Actual wins (optional): 112
Find: Expected winning percentage
Answer
10.
Problem 10.
- Runs (or points) scored: 700
- Runs (or points) allowed: 775
- Exponent: 1.83 (refined baseball)
- Games played: 130
- Actual wins (optional): 100
Find: Expected winning percentage
Answer

Answer key: Pythagorean Expectation worksheet

1. Expected winning percentage: 0.552
1. Here's the plan: raise both totals to the power k = 2. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{2}}{{RS}^{2} + {RA}^{2}}$
2. Raise runs scored to the exponent
  $750^{2}$ $= 562,500$
3. Do the same for runs allowed
  $675^{2}$ $= 455,625$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{562,500}{562,500 + 455,625}$ $= 0.552486$
5. Multiply the expected percentage by games played to get expected wins
  $0.552486 \times 190$ $= 104.972376$
6. Subtract expected wins from actual wins to see how far the real record is off
  $70 - 104.972376$ $= −34.972376$
2. Expected winning percentage: 0.633
1. Here's the plan: raise both totals to the power k = 1.83. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{1.83}}{{RS}^{1.83} + {RA}^{1.83}}$
2. Raise runs scored to the exponent
  $740^{1.83}$ $= 178,112.36$
3. Do the same for runs allowed
  $550^{1.83}$ $= 103,481.74$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{178,112.36}{178,112.36 + 103,481.74}$ $= 0.632515$
5. Multiply the expected percentage by games played to get expected wins
  $0.632515 \times 140$ $= 88.552034$
6. Subtract expected wins from actual wins to see how far the real record is off
  $100 - 88.552034$ $= 11.447966$
3. Expected winning percentage: 0.735
1. Here's the plan: raise both totals to the power k = 2. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{2}}{{RS}^{2} + {RA}^{2}}$
2. Raise runs scored to the exponent
  $900^{2}$ $= 810,000$
3. Do the same for runs allowed
  $540^{2}$ $= 291,600$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{810,000}{810,000 + 291,600}$ $= 0.735294$
5. Multiply the expected percentage by games played to get expected wins
  $0.735294 \times 160$ $= 117.647059$
6. Subtract expected wins from actual wins to see how far the real record is off
  $110 - 117.647059$ $= −7.647059$
4. Expected winning percentage: 0.759
1. Here's the plan: raise both totals to the power k = 3. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{3}}{{RS}^{3} + {RA}^{3}}$
2. Raise runs scored to the exponent
  ${1,100}^{3}$ $= 1,331,000,000$
3. Do the same for runs allowed
  $750^{3}$ $= 421,875,000$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{1,331,000,000}{1,331,000,000 + 421,875,000}$ $= 0.759324$
5. Multiply the expected percentage by games played to get expected wins
  $0.759324 \times 180$ $= 136.678314$
6. Subtract expected wins from actual wins to see how far the real record is off
  $110 - 136.678314$ $= −26.678314$
5. Expected winning percentage: 0.543
1. Here's the plan: raise both totals to the power k = 2. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{2}}{{RS}^{2} + {RA}^{2}}$
2. Raise runs scored to the exponent
  $600^{2}$ $= 360,000$
3. Do the same for runs allowed
  $550^{2}$ $= 302,500$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{360,000}{360,000 + 302,500}$ $= 0.543396$
5. Multiply the expected percentage by games played to get expected wins
  $0.543396 \times 150$ $= 81.509434$
6. Subtract expected wins from actual wins to see how far the real record is off
  $80 - 81.509434$ $= −1.509434$
6. Expected winning percentage: 0.369
1. Here's the plan: raise both totals to the power k = 2. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{2}}{{RS}^{2} + {RA}^{2}}$
2. Raise runs scored to the exponent
  $520^{2}$ $= 270,400$
3. Do the same for runs allowed
  $680^{2}$ $= 462,400$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{270,400}{270,400 + 462,400}$ $= 0.368996$
5. Multiply the expected percentage by games played to get expected wins
  $0.368996 \times 150$ $= 55.349345$
6. Subtract expected wins from actual wins to see how far the real record is off
  $120 - 55.349345$ $= 64.650655$
7. Expected winning percentage: 0.483
1. Here's the plan: raise both totals to the power k = 1.83. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{1.83}}{{RS}^{1.83} + {RA}^{1.83}}$
2. Raise runs scored to the exponent
  $675^{1.83}$ $= 150,530.99$
3. Do the same for runs allowed
  $700^{1.83}$ $= 160,890.13$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{150,530.99}{150,530.99 + 160,890.13}$ $= 0.483368$
5. Multiply the expected percentage by games played to get expected wins
  $0.483368 \times 220$ $= 106.340947$
6. Subtract expected wins from actual wins to see how far the real record is off
  $65 - 106.340947$ $= −41.340947$
8. Expected winning percentage: 0.645
1. Here's the plan: raise both totals to the power k = 1.83. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{1.83}}{{RS}^{1.83} + {RA}^{1.83}}$
2. Raise runs scored to the exponent
  $900^{1.83}$ $= 254,837.77$
3. Do the same for runs allowed
  $650^{1.83}$ $= 140,485.48$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{254,837.77}{254,837.77 + 140,485.48}$ $= 0.644631$
5. Multiply the expected percentage by games played to get expected wins
  $0.644631 \times 120$ $= 77.355765$
6. Subtract expected wins from actual wins to see how far the real record is off
  $88 - 77.355765$ $= 10.644235$
9. Expected winning percentage: 0.466
1. Here's the plan: raise both totals to the power k = 1.83. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{1.83}}{{RS}^{1.83} + {RA}^{1.83}}$
2. Raise runs scored to the exponent
  $720^{1.83}$ $= 169,401.97$
3. Do the same for runs allowed
  $775^{1.83}$ $= 193,830.5$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{169,401.97}{169,401.97 + 193,830.5}$ $= 0.466373$
5. Multiply the expected percentage by games played to get expected wins
  $0.466373 \times 160$ $= 74.619749$
6. Subtract expected wins from actual wins to see how far the real record is off
  $112 - 74.619749$ $= 37.380251$
10. Expected winning percentage: 0.454
1. Here's the plan: raise both totals to the power k = 1.83. The expected winning percentage is the runs-scored part's share of the two added together
  $pct$ $= \frac{{RS}^{1.83}}{{RS}^{1.83} + {RA}^{1.83}}$
2. Raise runs scored to the exponent
  $700^{1.83}$ $= 160,890.13$
3. Do the same for runs allowed
  $775^{1.83}$ $= 193,830.5$
4. Divide the runs-scored result by the two results added together. That gives the share of games the team should win
  $\frac{160,890.13}{160,890.13 + 193,830.5}$ $= 0.453569$
5. Multiply the expected percentage by games played to get expected wins
  $0.453569 \times 130$ $= 58.963915$
6. Subtract expected wins from actual wins to see how far the real record is off
  $100 - 58.963915$ $= 41.036085$