Answer:
Explanation:
Here are the answers to the tasks you listed:
**Miguel's game**
* **Xi** | **2** | **-1** | **P(xi)** |
|---|---|---|---|
| **2 chips with the same number** | 2 | 1/6 |
| **2 different numbers** | -1 | 5/6 |
* The expected value from playing the game is $0.167$.
* Miguel should expect to win or lose $0.167$ each time he plays.
* To make the game fair, the value of choosing two chips with the number 1 should be $-0.167$. This means that the expected value of the game would be $0$, which is fair.
**The wheel game**
* **Xi** | **3** | **1** | **-1** | **P(xi)** |
|---|---|---|---|
| **Landing on the blue sector** | 3 | 1/7 |
| **Landing on a yellow sector** | 1 | 2/7 |
| **Landing on a purple sector** | 0 | 2/7 |
| **Landing on a red sector** | -1 | 2/7 |
* The expected value from taking one spin is $0.142$.
* To make the game fair, the value of landing on the blue sector should be $-0.142$. This means that the expected value of the game would be $0$, which is fair.
**The basketball player**
* **Xi** | **3** | **0** | **P(xi)** |
|---|---|---|
| **Makes a three-point shot** | 3 | 0.30 |
| **Passes the ball** | 0 | 0.70 |
* The expected value of shooting the three-point shot is $0.90$.
* The expected value of passing the ball is $0$.
* Therefore, the basketball player should shoot the three-point shot himself.
**Claire's investment**
* The expected value of investing in the new business is $1,200$.
* Claire should invest in the company because the expected value is positive.
* It will take Claire approximately 1.2 years to earn back her initial investment.
**Tanya's games**
* **Xi** | **Lose $2** | **Win $1** | **Win $4** | **P(xi)** |
|---|---|---|---|
| **Game 1** | -2 | 0.20 | 0.25 | 0.55 |
| **Game 2** | -2 | 0.35 | 0.50 | 0.15 |
| **Game 3** | -2 | 0.60 | 0.20 | 0.20 |
* The expected value of playing Game 1 is $-0.10$.
* The expected value of playing Game 2 is $0.15$.
* The expected value of playing Game 3 is $0.00$.
* Therefore, Tanya should play Game 2.