A day trader buys an option on a stock that will return \$100 profit if the stock goes up today and lose \$200 if it goes down. If the trader thinks there is a 75% chance that the stock will go up, what is his expected value of the option?
How do I put this in the expectation equation?
Think of 75% as the probability that the stock goes up, i.e. $0.75$.
Then, the trader gains 100 with probability $0.75$, loses 200 with probability $1-0.75=0.25$. On expectation, what is his gain?
More formally: let $X$ be the random variable representing his gain. Then, $\mathbb{P}\{X=100\} = 0.75$, and $\mathbb{P}\{X=-100\} = 0.25$. You are asked to compute $\mathbb{E}X = \sum_{x}x\mathbb{P}\{X=x\}$.