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Data Interview Question

Choose the Perfect Prediction or Secure a Single Win

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Solution & Explanation

When faced with the choice between predicting all match outcomes correctly for a 1,000,000prizeorpredictingatleastonematchoutcomecorrectlyfora1,000,000 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(1/2)^7 = 1128\frac{1}{128}.
  • Expected Value:
    • Expected value for this scenario: 1,000,0001128=7,812.51,000,000 * \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(1/2)^7.
    • Therefore, probability of getting at least one match correct: 1(1/2)71 - (1/2)^7 = 111281 - \frac{1}{128} = 127128\frac{127}{128}.
  • Expected Value:
    • Expected value for this scenario: 10,000127128=9,921.87510,000 * \frac{127}{128} = 9,921.875.

Conclusion

  • The expected value for predicting at least one match correctly (9,921.875)ishigherthantheexpectedvalueforpredictingallmatchescorrectly(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.