Consider a game where you roll a six sided die and pick a card from a stack that contains three cards: a J, Q, and K. If you pay $1, you will win $5 if you pick a Jack and roll an even number. What is the expected value of this game? Round to two decimal places
Consider a game where you roll a six sided die and pick a card from a stack that contains three cards: a J, Q, and K. If you pay $1, you will win $5 if you pick a Jack and roll an even number. What is the expected value of this game? Round to two decimal places
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Game Description:**
Consider a game where you roll a six-sided die and pick a card from a stack that contains three cards: a Jack (J), a Queen (Q), and a King (K). If you pay $1, you will win $5 if you pick a Jack and roll an even number.
**Objective:**
Calculate the expected value of this game. Round to two decimal places.
**Solution:**
1. **Probability of Rolling an Even Number:**
- The even numbers on a six-sided die are 2, 4, and 6.
- Thus, the probability of rolling an even number is \( \frac{3}{6} = \frac{1}{2} \).
2. **Probability of Picking a Jack:**
- The stack contains three cards: J, Q, K.
- Thus, the probability of picking a Jack is \( \frac{1}{3} \).
3. **Combined Probability:**
- Probability of both picking a Jack and rolling an even number is \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).
4. **Expected Value Calculation:**
- The gain if you win is $5, but you initially pay $1 to play, so the net gain is $4.
- The expected value (EV) is calculated as:
\[
EV = \left(\frac{1}{6} \times 4\right) + \left(\frac{5}{6} \times (-1)\right)
\]
\[
EV = \frac{4}{6} - \frac{5}{6} = -\frac{1}{6} \approx -0.17
\]
**Expected Value:**
The expected value of the game is approximately \(-\$0.17\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dbed1da-3f32-4254-ad4d-67b595b99f21%2F4e595298-1e6a-45b2-96dd-8ee2169920c9%2Fhepxab_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Game Description:**
Consider a game where you roll a six-sided die and pick a card from a stack that contains three cards: a Jack (J), a Queen (Q), and a King (K). If you pay $1, you will win $5 if you pick a Jack and roll an even number.
**Objective:**
Calculate the expected value of this game. Round to two decimal places.
**Solution:**
1. **Probability of Rolling an Even Number:**
- The even numbers on a six-sided die are 2, 4, and 6.
- Thus, the probability of rolling an even number is \( \frac{3}{6} = \frac{1}{2} \).
2. **Probability of Picking a Jack:**
- The stack contains three cards: J, Q, K.
- Thus, the probability of picking a Jack is \( \frac{1}{3} \).
3. **Combined Probability:**
- Probability of both picking a Jack and rolling an even number is \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).
4. **Expected Value Calculation:**
- The gain if you win is $5, but you initially pay $1 to play, so the net gain is $4.
- The expected value (EV) is calculated as:
\[
EV = \left(\frac{1}{6} \times 4\right) + \left(\frac{5}{6} \times (-1)\right)
\]
\[
EV = \frac{4}{6} - \frac{5}{6} = -\frac{1}{6} \approx -0.17
\]
**Expected Value:**
The expected value of the game is approximately \(-\$0.17\).
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