Calculate the expected value E of a random variable W . Identify the possible outcomes of random variable $$ W $$ with respective Wrobabilities of occurring $$ P(W) $$ simplify. P(red) $$ =\frac{9}{19} $$ Multiply and determine the anticipated result. $$ -\$ 0.09 imes 7=-\$ 0.63 $$ Use the Complement Rule of Probability of the ball landing on a red pocket; make sure to the probability of not Ianding on a red pocket. Identify the key information: There are 38 possible pockets on the wheel, two green pockets containing zeros, and 18 red pockets.

Question

Calculate the expected value E of a random variable W . Identify the possible outcomes of random variable $$ W $$ with respective Wrobabilities of occurring $$ P(W) $$ simplify. P(red) $$ =\frac{9}{19} $$ Multiply and determine the anticipated result. $$ -\$ 0.09 	imes 7=-\$ 0.63 $$ Use the Complement Rule of Probability of the ball landing on a red pocket; make sure to the probability of not Ianding on a red pocket. Identify the key information: There are 38 possible pockets on the wheel, two green pockets containing zeros, and 18 red pockets.

UpStudy AI · Accepted Answer

To calculate the expected value $$ E $$ of the random variable $$ W $$, we need to follow these steps:

1. **Identify the possible outcomes and their probabilities**:
   - The outcomes are based on the color of the pocket where the ball lands.
   - There are 38 pockets in total: 18 red, 20 non-red (which includes 2 green and 18 black).

2. **Calculate the probabilities**:
   - The probability of landing on a red pocket is given as $$ P(\text{red}) = \frac{9}{19} $$.
   - The probability of not landing on a red pocket (using the complement rule) is:
     $$
     P(\text{not red}) = 1 - P(\text{red}) = 1 - \frac{9}{19} = \frac{10}{19}
     $$

3. **Define the outcomes for the random variable $$ W $$**:
   - Let $$ W $$ represent the winnings based on the outcome:
     - If the ball lands on red, the outcome is $$ W = -0.09 $$ (loss).
     - If the ball does not land on red, the outcome is $$ W = 0 $$ (no loss).

4. **Calculate the expected value $$ E(W) $$**:
   - The expected value is calculated as follows:
   $$
   E(W) = P(\text{red}) \cdot W_{\text{red}} + P(\text{not red}) \cdot W_{\text{not red}}
   $$
   - Substituting the values:
   $$
   E(W) = \left(\frac{9}{19} \cdot (-0.09)\right) + \left(\frac{10}{19} \cdot 0\right)
   $$

Now, let's perform the calculation for $$ E(W) $$.

### Calculation
We will calculate $$ E(W) $$ using the expression derived above.

$$
E(W) = \frac{9}{19} \cdot (-0.09) + \frac{10}{19} \cdot 0
$$

This simplifies to:

$$
E(W) = \frac{9}{19} \cdot (-0.09)
$$

Now, let's calculate this value.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{9}{19}\left(-0.09\right)$$
- step1: Convert the expressions:
  $$\frac{9}{19}\left(-\frac{9}{100}\right)$$
- step2: Multiply the fractions:
  $$\frac{9\left(-9\right)}{19\times 100}$$
- step3: Multiply the terms:
  $$\frac{-81}{19\times 100}$$
- step4: Multiply the terms:
  $$\frac{-81}{1900}$$
- step5: Multiply the numbers:
  $$-\frac{81}{1900}$$
The expected value $$ E(W) $$ is calculated as follows:

$$
E(W) = \frac{9}{19} \cdot (-0.09) \approx -0.042632
$$

Thus, the expected value $$ E(W) $$ is approximately $$ -0.0426 $$.

### Summary of Key Information:
- Probability of landing on a red pocket: $$ P(\text{red}) = \frac{9}{19} $$
- Probability of not landing on a red pocket: $$ P(\text{not red}) = \frac{10}{19} $$
- Expected value $$ E(W) \approx -0.0426 $$

This means that, on average, you can expect to lose about $$ 0.0426 $$ dollars per game played under these conditions.
The expected value $$ E(W) $$ is approximately $$ -0.0426 $$ dollars.

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