1. Explain the probability distribution for each die.-calculate expected value and variance for each die - If you roll a 1, you get $1.- If you roll a 2, you get $2.- If you roll a 3, you get $0. (That's right: you get no money.)- If you roll a 4, you get $4.- If you roll a 5, you get $5.- If you roll a 6, you get $6. Scarlett has two six-sided dice: one red and one blue, both of which are loaded. A fair die has a probability of 1/6 for landing on any side, but these dice have specific biases.**The Red Die:**- Odd numbers are twice as likely as even numbers.- Numbers 1, 3, and 5 have equal likelihood.- Numbers 2, 4, and 6 have equal likelihood.**The Blue Die:**- Numbers 1 and 5 are equally likely.- Numbers 2 and 4 are equally likely.- Numbers 3 and 6 are equally likely.- Number 3 is twice as likely as number 2.- Number 2 is twice as likely as number 1.**Games:**Scarlett offers three games to play:1. **First Game:** - Choose a die (red or blue) and roll it once. - Rewards based on roll: - Roll a 1: Get $1 - Roll a 2: Get $2 - Roll a 3: Get $3 - Roll a 4: Get $4 - Roll a 5: Get $5 - Roll a 6: Get $62. **Second Game:** - Choose a die and roll it four times. - If any roll results in a 3, win $0. - If no roll results in a 3, win $3.3. **Third Game:** - Choose a die and roll it once. Only game rules mentioned, further instructions not provided.

Name: 1. Explain the probability distribution for each die.-calculate expect
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Description: 1. Explain the probability distribution for each die.-calculate expected value and variance for each die - If you roll a 1, you get $1.- If you roll a 2, you get $2.- If you roll a 3, you get $0. (That's right: you get no money.)- If you roll a 4, yo

Answered: Explain the probability distribution…

A First Course in Probability (10th Edition)

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ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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1. Explain the probability distribution for each die.

-calculate expected value and variance for each die

- If you roll a 1, you get $1.
- If you roll a 2, you get $2.
- If you roll a 3, you get $0. (That's right: you get no money.)
- If you roll a 4, you get $4.
- If you roll a 5, you get $5.
- If you roll a 6, you get $6.

Scarlett has two six-sided dice: one red and one blue, both of which are loaded. A fair die has a probability of 1/6 for landing on any side, but these dice have specific biases.

**The Red Die:**

- Odd numbers are twice as likely as even numbers.
- Numbers 1, 3, and 5 have equal likelihood.
- Numbers 2, 4, and 6 have equal likelihood.

**The Blue Die:**

- Numbers 1 and 5 are equally likely.
- Numbers 2 and 4 are equally likely.
- Numbers 3 and 6 are equally likely.
- Number 3 is twice as likely as number 2.
- Number 2 is twice as likely as number 1.

**Games:**

Scarlett offers three games to play:

1. **First Game:**
- Choose a die (red or blue) and roll it once.
- Rewards based on roll:
- Roll a 1: Get $1
- Roll a 2: Get $2
- Roll a 3: Get $3
- Roll a 4: Get $4
- Roll a 5: Get $5
- Roll a 6: Get $6

2. **Second Game:**
- Choose a die and roll it four times.
- If any roll results in a 3, win $0.
- If no roll results in a 3, win $3.

3. **Third Game:**
- Choose a die and roll it once. Only game rules mentioned, further instructions not provided.

Expert Solution

Step 1

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Let X : number occurs on the die

X={1,2,3,4,5,6}

Let Y : the prize money

Step 2

The probability distribution of X for each die is given by:

X	P(x)
1	$\frac{1}{6}$
2	$\frac{1}{6}$
3	$\frac{1}{6}$
4	$\frac{1}{6}$
5	$\frac{1}{6}$
6	$\frac{1}{6}$

The probability distribution of Y and expected Y is given as:

Y (in $)	P(Y)	E(Y) (in $)
1	$\frac{1}{6}$	$\frac{1}{6}$
2	$\frac{1}{6}$	$\frac{1}{3}$
0	$\frac{1}{6}$	$0$
4	$\frac{1}{6}$	$\frac{2}{3}$
5	$\frac{1}{6}$	$\frac{5}{6}$
6	$\frac{1}{6}$	$1$
Total	1	3

Where the expected prize money is given by :

$E (y) = \sum_{y} y \times P (y) = 3$

Therefore, the expected value for each die is given as:

Die	Expected Value (in $)
1
2
3	0
4
5
6	1
Total	3

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