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Question 17 of 22
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Suppose someone gives you 13 to 4 odds that you cannot roll two even numbers with the roll of two fair dice. This means you win $\$ 13$ if you succeed and you lose $\$ 4$ if you fail. What is the expected value of this game to you? Should you expect to win or lose the expected value in the first game? What can you expect if you play 200 times?
Explain. (The table will be helpful in finding the required probabilities.)
Click the icon to view the table.
What is the expected value of this game to you?
$\$ \square$
Should you expect to win (or lose) an amount equal to the expected value in the first game?
Yes, you can expect to win (or lose) the expected value in the first game.
No, the outcome of one game cannot be predicted.
What can you expect if you play 200 times?
\[
\$ \square
\]
Explain this result.
Averaged over 200 games, you should expect to $\square \$ \square$.
Answer
\(\boxed{\text{Final Answer: The expected value of this game to you is \$0.25. If you play 200 times, you can expect to win \$50.}}\)
Step 1 ：First, we need to calculate the probability of rolling two even numbers. The probability of winning is 1/4 and the probability of losing is 3/4.
Step 2 ：Next, we calculate the expected value of the game. The expected value is the sum of the possible outcomes each multiplied by their respective probabilities. In this case, the expected value is \(0.25 \times 13 + 0.75 \times -4 = \$0.25\).
Step 3 ：We should not expect to win or lose an amount equal to the expected value in the first game, as the outcome of one game cannot be predicted.
Step 4 ：If we play 200 times, we can calculate the expected value over 200 games by multiplying the expected value by 200. This gives us \(0.25 \times 200 = \$50\).
Step 5 ：Averaged over 200 games, we should expect to win \$50.
Step 6 ：\(\boxed{\text{Final Answer: The expected value of this game to you is \$0.25. If you play 200 times, you can expect to win \$50.}}\)
