We have the following discrete random: If the result of throwing a perfect dice is a prime number, the payoff will be the result times $$10$$. We include in the table these payoffs. Assign payoffs to the other results from throwing the dice.
 Fill in the following table:
Result of the dice  probability  payoff 
$$1$$  $$1/6$$  $$10$$ 
$$2$$  ?  ? 
$$3$$  ?  $$30$$ 
$$4$$  ?  ? 
$$5$$  $$1/6$$  ? 
$$6$$  $$1/6$$  ? 

Find the average payoff if we throw the dice only once.
 Find the variance and the standard deviation.
Development:
Result of the dice  probability  payoff 
$$1$$  $$1/6$$  $$10$$ 
$$2$$  $$1/6$$  $$20$$ 
$$3$$  $$1/6$$  $$30$$ 
$$4$$  $$1/6$$  $$8$$ 
$$5$$  $$1/6$$  $$50$$ 
$$6$$  $$1/6$$  $$120$$ 

$$$\mu=\sum_i p_i\cdot x_i=\dfrac{1}{6}\cdot10+\dfrac{1}{6}\cdot20+\dfrac{1}{6}\cdot30+\dfrac{1}{6}\cdot8+\dfrac{1}{6}\cdot50+\dfrac{1}{6}\cdot120$$$ $$$\mu=\dfrac{238}{6}=39,67$$$

The variance is calculated first: $$$\sigma^2=\sum_i x_i^2\cdot p_i  \mu^2=\dfrac{1}{6}(10^2+20^2+30^2+8^2+50^2+120^2)39,67^2$$$
variance $$\rightarrow \sigma^2=1486,95$$
standard deviation $$\rightarrow \sigma=38,56$$
Solution:
Result of the dice  probability  payoff 
$$1$$  $$1/6$$  $$10$$ 
$$2$$  $$1/6$$  $$20$$ 
$$3$$  $$1/6$$  $$30$$ 
$$4$$  $$1/6$$  $$8$$ 
$$5$$  $$1/6$$  $$50$$ 
$$6$$  $$1/6$$  $$120$$ 

$$\mu=\dfrac{238}{6}=39,67$$

variance $$\rightarrow \sigma^2=1486,95$$
standard deviation $$\rightarrow \sigma=38,56$$