You are on a TV show. You have been asked to either play a dice game five times or accept a $50 bill. The dice game works like this:
If you roll a 1, 2 or 3, you win $46.
If you roll a 4 or 5, you lose $22.
If you roll a 6, you lose $85.
Should you play the game? Use expected values and decision theory to justify your answer. (Round the answer to the nearest cent.)
Take the $50 and quit.
(Each game as outlined by drwls has an expected value of $1.50.
You are playing it 5 times, so the expected return is $7.50.
Your choice was to either accept $50 or play the game)
Reiny is correct. I assumed you were betting your own money. I didn't read all the question.
To determine whether you should play the dice game or accept the $50 bill, we need to compare the expected values of the two options. The expected value is a measure of the average outcome of a random event.
Let's calculate the expected value for playing the dice game.
The probability of rolling a 1, 2, or 3 is 3/6 = 1/2.
The probability of rolling a 4 or 5 is 2/6 = 1/3.
The probability of rolling a 6 is 1/6.
The corresponding gains/losses for each outcome are as follows:
Rolling a 1, 2, or 3: +$46
Rolling a 4 or 5: -$22
Rolling a 6: -$85
Now, we can calculate the expected value:
Expected Value = (Probability of rolling a 1, 2, or 3 * Gain/Loss for rolling a 1, 2, or 3)
+ (Probability of rolling a 4 or 5 * Gain/Loss for rolling a 4 or 5)
+ (Probability of rolling a 6 * Gain/Loss for rolling a 6)
Expected Value = (1/2 * $46) + (1/3 * -$22) + (1/6 * -$85)
Simplifying the calculation, we have:
Expected Value = $23 - $7.33 - $14.17
Expected Value = $1.50 (rounded to the nearest cent)
The expected value for playing the dice game is $1.50.
Now, let's consider the alternative option of accepting the $50 bill, which has a fixed value.
Comparing the expected value of playing the dice game ($1.50) to the fixed value of the $50 bill, we can see that the expected value is significantly lower than $50.
Based on decision theory, which suggests selecting the option with the highest expected value, it is more favorable to accept the $50 bill instead of playing the dice game.
Therefore, you should choose to accept the $50 bill rather than playing the dice game.
The expected winning per game is
(1/2)*46 - (1/3)*22 - (1/6)*85
If that is positive, play the game