Dice Game Profitability

Solution & Explanation

To determine whether the dice game is profitable, we need to calculate the expected value of playing the game. The expected value is the average amount of money you can expect to win or lose per game if you play the game many times.

Step 1: Determine the Probability of Winning

• Total Possible Outcomes:

  • Each die has 6 sides, so there are a total of 6 x 6 = 36 possible outcomes when rolling two dice.

• Favorable Outcomes:

  • To win, the sum of the dice must be 7. The possible combinations for this are:
    • (1, 6)
    • (2, 5)
    • (3, 4)
    • (4, 3)
    • (5, 2)
    • (6, 1)
  • There are 6 favorable outcomes.

• Probability of Winning:

  • The probability of rolling a sum of 7 is 6/36 = 1/6.

Step 2: Calculate the Expected Value

• Game Costs and Winnings:

  • You win $21 if the sum is 7.
  • You lose $10 (the cost of playing) if the sum is not 7.

• Expected Value Calculation:

  • Let X be the profit from a single game.
  • The possible values of X are:
    • $11 if you win ($21 winnings - $10 cost)
    • -$10 if you lose (since you pay$10 and win nothing)

• Expected Value Formula:

  $E[X] = (\text{Probability of Winning} \times \text{Profit if Win}) + (\text{Probability of Losing} \times \text{Loss if Lose})$

  $E[X] = \left(\frac{1}{6} \times 11\right) + \left(\frac{5}{6} \times (-10)\right)$

  $E[X] = \frac{11}{6} - \frac{50}{6} = -\frac{39}{6} = -6.5$

Conclusion

The expected value of the game is -$6.50. This means that, on average, you would lose$6.50 every time you play the game. Therefore, from a purely mathematical standpoint, this game is not profitable and is not worth playing if your goal is to make money.

Additional Considerations

• Risk and Entertainment:

  • While the game is not profitable, some may still choose to play for entertainment value or the thrill of gambling.

• Long-term vs Short-term:

  • In the short term, you might get lucky and win a few games, but over many games, the expected losses will likely manifest.

• Financial Implications:

  • It's crucial to consider your financial situation and whether you can afford potential losses before deciding to engage in such games.