Suppose you toss three fair coins. You win 32 cents if all three coins come up the same.
You win 8 cents if exactly 1 head occurs, and you win 48 cents if exactly 2 heads appears.
Would you pay 30 cents to play this game?
If not, then how much would you be willing to pay?
To determine whether it is worth paying 30 cents to play this game, we need to calculate the expected value. The expected value is the average amount you can expect to win per game.
Let's calculate the probability of each outcome first:
1. All three coins come up the same (either all heads or all tails): There are two possibilities (HHH or TTT), and each has a probability of 1/2 * 1/2 * 1/2 = 1/8. Winning amount = 32 cents.
2. Exactly 1 head occurs: There are three possibilities (HHT, HTH, THH), and each has a probability of 1/2 * 1/2 * 1/2 = 1/8. Winning amount = 8 cents.
3. Exactly 2 heads appear: There are three possibilities (HTT, THT, TTH), and each has a probability of 1/2 * 1/2 * 1/2 = 1/8. Winning amount = 48 cents.
Now let's calculate the expected value:
Expected value = (Probability of Outcome 1 * Winning Amount for Outcome 1) +
(Probability of Outcome 2 * Winning Amount for Outcome 2) +
(Probability of Outcome 3 * Winning Amount for Outcome 3)
Expected value = (1/8 * 32) + (1/8 * 8) + (1/8 * 48)
Expected value = 4 + 1 + 6
Expected value = 11 cents
The expected value for this game is 11 cents. In other words, on average, you can expect to win 11 cents per game.
Since the expected value per game is positive (greater than 0), it means that, in the long run, you would be expected to make a profit by playing this game. Therefore, it would be worth paying 30 cents to play this game.
However, if you want to determine how much you would be willing to pay to play this game, you need to decide on your own personal threshold for what constitutes a worthwhile investment.