Question

Consider the effect of a change in the game so that if tails comes up two times in a row, you get nothing. How would your answers to the first three parts of this question change? Possibilities Probability Outcome Payoff 1 8 / 27 3 tails, 0 heads $ 0 2 4 / 27 tails, heads, tails $ 100 3 4 / 27 tails, tails, heads $ 0 4 4 / 27 heads, tails, tails $ 0 5 2 / 9 2 heads, 1 tails $ 200 6 1 / 27 3 heads, 0 tails $ 300 Expected value = $ 200

Answer 1

Answer: Expected Value ≈ $70.36

Explanation:

If the game is modified so that getting two tails in a row results in getting nothing, the expected value would change as follows for each possibility:

1. If you get three tails in a row (Outcome 1), you get $0, as in the previous scenario.

2. If you get tails, heads, tails (Outcome 2), you get $100, just like in the previous scenario.

3. If you get tails, tails, heads (Outcome 3), you get $0, but this time, it's due to the two tails in a row rule.

4. If you get heads, tails, tails (Outcome 4), you get $0, similar to the previous scenario.

5. If you get two heads and one tail (Outcome 5), you get $200, similar to the previous scenario.

6. If you get three heads in a row (Outcome 6), you get $300, just like in the previous scenario.

Now, let's calculate the new expected value:

Expected Value = (Probability of Outcome 1 * Payoff for Outcome 1) + (Probability of Outcome 2 * Payoff for Outcome 2) + (Probability of Outcome 3 * Payoff for Outcome 3) + (Probability of Outcome 4 * Payoff for Outcome 4) + (Probability of Outcome 5 * Payoff for Outcome 5) + (Probability of Outcome 6 * Payoff for Outcome 6)

Expected Value = (8/27 * $0) + (4/27 * $100) + (4/27 * $0) + (4/27 * $0) + (2/9 * $200) + (1/27 * $300)

Expected Value = $0 + $14.81 + $0 + $0 + $44.44 + $11.11

Expected Value ≈ $70.36

So, with the modification to the game rules, the new expected value is approximately $70.36. This is different from the previous expected value of $200 due to the possibility of getting nothing when two tails come up in a row.