Calculating the expected value of rolling a dice is actually pretty simple! Let’s break it down so it’s easy to understand.

First, a regular die has six sides, numbered from 1 to 6. When you roll it, any of these numbers has the same chance of coming up. The chance of rolling any particular number is equal, which is $1/6$.

To find the expected value, or EV for short, we use this straightforward formula:

$$EV = (P_1 \times X_1) + (P_2 \times X_2) + (P_3 \times X_3) + (P_4 \times X_4) + (P_5 \times X_5) + (P_6 \times X_6)$$

Here’s what the symbols mean:

• $P_n$ is the chance of rolling a specific number.
• $X_n$ is the number you rolled.

Since every number has the same chance ($P_n = \frac{1}{6}$ for the numbers 1 to 6), we can make our math a little easier.

Let’s figure out the expected value step by step:

1. For rolling a 1: $P_1 \times X_1 = \frac{1}{6} \times 1 = \frac{1}{6}$
2. For rolling a 2: $P_2 \times X_2 = \frac{1}{6} \times 2 = \frac{2}{6}$
3. For rolling a 3: $P_3 \times X_3 = \frac{1}{6} \times 3 = \frac{3}{6}$
4. For rolling a 4: $P_4 \times X_4 = \frac{1}{6} \times 4 = \frac{4}{6}$
5. For rolling a 5: $P_5 \times X_5 = \frac{1}{6} \times 5 = \frac{5}{6}$
6. For rolling a 6: $P_6 \times X_6 = \frac{1}{6} \times 6 = \frac{6}{6}$

Now, let’s add these all together:

$$EV = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6} = \frac{21}{6} = 3.5$$

So, the expected value for rolling a dice one time is $3.5$.

This means that if you roll the die many times, you can expect the average result to be around $3.5$. Remember, it's not a number you will actually roll, but it gives us a good idea of what to expect in the long run!