Juan is playing a game in which he spins a spinner with 6 equal-sized slices numbered 1 through 6 . The spinner stops on a numbered slice at random.
This game is this: Juan spins the spinner once. He wins if the spinner stops on the number if the spinner stops on the number 2 , if the spinne
stops on the number 3 , and if the spinner stops on the number 4 . He loses if the spinner stops on 5 or 6 .
(a) Find the expected value of playing the game.
(b) What can Juan expect in the long run, after playing the game many times?
Juan can expect to gain money.
He can expect to win dollars per spin.
He can expect to lose dollars per spin.
O Juan can expect to break even (neither gain nor lose money).

Ask by Sherman Jimenez. in the United States

Jan 06,2025

Question

Juan is playing a game in which he spins a spinner with 6 equal-sized slices numbered 1 through 6 . The spinner stops on a numbered slice at random.
This game is this: Juan spins the spinner once. He wins if the spinner stops on the number if the spinner stops on the number 2 , if the spinne
stops on the number 3 , and if the spinner stops on the number 4 . He loses if the spinner stops on 5 or 6 .
(a) Find the expected value of playing the game.
(b) What can Juan expect in the long run, after playing the game many times?
Juan can expect to gain money.
He can expect to win dollars per spin.
He can expect to lose dollars per spin.
O Juan can expect to break even (neither gain nor lose money).

Ask by Sherman Jimenez. in the United States

Jan 06,2025

UpStudy AI · Accepted Answer

Let's analyze Juan's game step by step.

### Part (a): Finding the Expected Value

The **expected value (E)** of a game is the average amount Juan can expect to win or lose per spin if he plays the game many times. It is calculated by multiplying each possible outcome by its probability and then summing all these values.

Here are the possible outcomes and their corresponding payouts:

- **Spin stops on 1:** Win \$1
- **Spin stops on 2:** Win \$4
- **Spin stops on 3:** Win \$7
- **Spin stops on 4:** Win \$10
- **Spin stops on 5 or 6:** Lose \$11 each

Since the spinner has 6 equal-sized slices, the probability of landing on any particular number is $$ \frac{1}{6} $$.

Now, calculate the expected value:

$$
E = \left( \frac{1}{6} \times 1 \right) + \left( \frac{1}{6} \times 4 \right) + \left( \frac{1}{6} \times 7 \right) + \left( \frac{1}{6} \times 10 \right) + \left( \frac{1}{6} \times (-11) \right) + \left( \frac{1}{6} \times (-11) \right)
$$

$$
E = \frac{1 + 4 + 7 + 10 - 11 - 11}{6} = \frac{0}{6} = 0
$$

So, the **expected value is \$0**.

### Part (b): Long-Run Expectation

Since the expected value is \$0, this means that **on average, Juan neither gains nor loses money** per spin in the long run. Over many spins, the gains and losses balance each other out.

**Summary:**
- **Expected Value:** \$0
- **Long-Run Expectation:** Juan can expect to break even (neither gain nor lose money) per spin.

**Answer:**

Problem (b) Answer:

Juan can expect to break even (neither gain nor lose money).
Juan can expect to break even (neither gain nor lose money) per spin.

Upstudy AI Solution

Answer

Solution

Part (a): Finding the Expected Value

Part (b): Long-Run Expectation

Mind Expander

Related Questions

Latest Probability Questions