Choose the Perfect Prediction or Secure a Single Win

Solution & Explanation

When faced with the choice between predicting all match outcomes correctly for a $1,000,000 prize or predicting at least one match outcome correctly for a$10,000 prize, we need to consider the probabilities and expected values of each scenario.

1. Predicting All Match Outcomes Correctly

• Scenario: Predict all the match outcomes in a tournament of 8 teams correctly.
• Total Matches: There are 7 matches in total (3 rounds: 4 quarter-finals, 2 semi-finals, 1 final).
• Probability Calculation:
  • Each match has a probability of 1/2 for being predicted correctly.
  • The probability of predicting all matches correctly: $(1/2)^7$ = $\frac{1}{128}$.
• Expected Value:
  • Expected value for this scenario: $1,000,000 * \frac{1}{128} = 7,812.5$.

2. Predicting At Least One Match Correctly

• Scenario: Predict at least one match correctly out of the 7 matches.
• Probability Calculation:
  • Probability of predicting at least one match correctly is the complement of predicting all matches incorrectly.
  • Probability of getting all matches wrong: $(1/2)^7$.
  • Therefore, probability of getting at least one match correct: $1 - (1/2)^7$ = $1 - \frac{1}{128}$ = $\frac{127}{128}$.
• Expected Value:
  • Expected value for this scenario: $10,000 * \frac{127}{128} = 9,921.875$.

Conclusion

• The expected value for predicting at least one match correctly ($9,921.875) is higher than the expected value for predicting all matches correctly ($7,812.5).
• Therefore, it is statistically more beneficial to choose the option of receiving $10,000 for correctly predicting at least one match outcome.

This analysis demonstrates the importance of considering both probability and expected value when making decisions under uncertainty, a crucial skill in data science and analytical roles.