Maria is playing a game of chance in which she rolls a number cube with sides numbered from 1 to 6 .

/ Exams & Certification / By

Maria is playing a game of chance in which she rolls a number cube with sides numbered from 1 to 6 . The number cube is fair, so a side is rolled at random. This game is this: Maria rolls the number cube once. She wins $1 if a 1 is rolled, $2 if a 2 is rolled, $3 if a 3 is rolled, and $4 if a 4 is rolled. She loses $2.75 if a 5 or 6 is rolled.

The Correct Answer and Explanation is:

To calculate the expected value of Maria’s game, we need to account for all possible outcomes and their corresponding probabilities.

The game has 6 possible outcomes, one for each side of the number cube, each with an equal probability of 16\frac{1}{6}61​. The outcomes and their associated winnings (or losses) are as follows:

  • Rolling a 1: Maria wins $1.
  • Rolling a 2: Maria wins $2.
  • Rolling a 3: Maria wins $3.
  • Rolling a 4: Maria wins $4.
  • Rolling a 5 or 6: Maria loses $2.75 (each of these two outcomes has a probability of 16\frac{1}{6}61​).

Step 1: Calculate the expected value

The expected value (EV) is a weighted average of all possible outcomes, where each outcome is multiplied by its probability. The formula for expected value is:EV=∑(outcome×probability of that outcome)EV = \sum (\text{outcome} \times \text{probability of that outcome})EV=∑(outcome×probability of that outcome)

Let’s calculate it for each outcome:

  • For rolling a 1:

1×16=161 \times \frac{1}{6} = \frac{1}{6}1×61​=61​

  • For rolling a 2:

2×16=262 \times \frac{1}{6} = \frac{2}{6}2×61​=62​

  • For rolling a 3:

3×16=363 \times \frac{1}{6} = \frac{3}{6}3×61​=63​

  • For rolling a 4:

4×16=464 \times \frac{1}{6} = \frac{4}{6}4×61​=64​

  • For rolling a 5:

−2.75×16=−2.756-2.75 \times \frac{1}{6} = -\frac{2.75}{6}−2.75×61​=−62.75​

  • For rolling a 6:

−2.75×16=−2.756-2.75 \times \frac{1}{6} = -\frac{2.75}{6}−2.75×61​=−62.75​

Step 2: Sum all the contributions

Now, let’s add these up to find the total expected value:EV=16+26+36+46+(−2.756)+(−2.756)EV = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \left(-\frac{2.75}{6}\right) + \left(-\frac{2.75}{6}\right)EV=61​+62​+63​+64​+(−62.75​)+(−62.75​)

Simplifying:EV=1+2+3+4−2.75−2.756EV = \frac{1 + 2 + 3 + 4 – 2.75 – 2.75}{6}EV=61+2+3+4−2.75−2.75​EV=7.56=1.25EV = \frac{7.5}{6} = 1.25EV=67.5​=1.25

Conclusion:

The expected value of Maria’s game is $1.25. This means that, on average, Maria can expect to win $1.25 each time she plays the game.

Related Posts

Scroll to Top