Chapter 17: Making Decisions With Uncertainty 17-2 Game Show ✓ Solved

Chapter 17: Making Decisions With Uncertainty 17-2 Game Show Uncertainty

The final round of a TV game show offers contestants the opportunity to increase their current winnings of $1 million to $2 million. If the contestant's guess is correct, they win $2 million; if incorrect, their winnings decrease to $500,000. The contestant believes their probability of being correct is 50%. The question is whether the contestant should play. Additionally, the task is to determine the lowest probability of a correct guess that would make playing a profitable decision.

Analysis of the Game Show Decision

To decide if the contestant should play, we analyze the expected value (EV) of playing the game. The EV incorporates the probabilities of winning or losing and the respective payoffs.

Given:

  • Current winnings: $1,000,000
  • Winnings if correct: $2,000,000
  • Winnings if incorrect: $500,000
  • Probability of correctness: 50% or 0.5

Calculating EV:

EV = (Probability of correct) × (Payoff if correct) + (Probability of incorrect) × (Payoff if incorrect)

EV = 0.5 × $2,000,000 + 0.5 × $500,000 = $1,000,000 + $250,000 = $1,250,000

The expected winnings after playing is $1,250,000, which exceeds the current winnings of $1,000,000. Therefore, from a purely expected value perspective, the contestant should play.

Next, to determine the lowest probability at which playing remains profitable, we set the expected value equal to the current winnings ($1,000,000) and solve for the probability (p):

EV = p × $2,000,000 + (1 - p) × $500,000 ≥ $1,000,000

p × $2,000,000 + (1 - p) × $500,000 = $1,000,000

Expand:

$2,000,000 p + $500,000 - $500,000 p = $1,000,000

Simplify:

($2,000,000 p - $500,000 p) + $500,000 = $1,000,000

($1,500,000 p) + $500,000 = $1,000,000

$1,500,000 p = $1,000,000 - $500,000 = $500,000

p = $500,000 / $1,500,000 ≈ 0.3333 or 33.33%

Thus, the lowest probability of a correct guess at which playing is profitable is approximately 33.33%. Below this threshold, the expected value would be less than the current winnings, making it unprofitable to play.

Conclusion

The contestant should play the game because the expected value at a 50% chance of success exceeds the current winnings. The lowest probability of being correct that justifies playing is approximately 33.33%. If the contestant believes the probability of being correct is at least 33.33%, then, from a decision-theoretic standpoint, playing the game is the rational choice.

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