Find the expected value of the number on a dice when thrown.
Let $X$ be a random variable which represents the value of a number when a dice is thrown.
Then $X$ takes values from $1$ to $6$.
P(getting any given number on the dice) $= \dfrac{1}{6}$
$\therefore \;$ We have
| $X$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
|---|---|---|---|---|---|---|
| $P \left(X = x\right)$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ |
Expected value of $X = E \left(X\right) = \sum \limits_{i} x_i p_i$
$\begin{aligned} \therefore \; E \left(X\right) & = \left(1 \times \dfrac{1}{6}\right) + \left(2 \times \dfrac{1}{6}\right) + \left(3 \times \dfrac{1}{6}\right) + \left(4 \times \dfrac{1}{6}\right) + \left(5 \times \dfrac{1}{6}\right) + \left(6 \times \dfrac{1}{6}\right) \\\\ & = \dfrac{21}{6} = 3.5 \end{aligned}$