I'm not very good with math but I am trying to understand why it is 50/50. From what I am thinking, you have two options for a trade: buy or short-sell. That is 50/50. But for whichever option you choose, you have two outcomes: price goes up or price goes down. Would that be 1/2 * 1/2 = 1/4? That is, there is a 25% chance that price will go up and a 25% chance that price will go down for every trade. But then, if I take the difference (75% chance), then that would mean one can win 75% of the time! And that doesn't make sense. What am I missing/not understanding?
You buy the stock, 50% it goes up and you win, 50% it goes down and you lose. You sell it short, 50% it goes up and you lose, 50% and it goes down and you win. Each strategy gives you 50% chance of winning. So if you choose your strategy by flipping a fair coin (not sure why you would do this) 1/2*1/2 + 1/2*1/2= 1/4+1/4 = 1/2.
I'd be careful tossing around the word "options" like that since it has a relevant meaning in investing that is different from what I think you're saying, which is "options" as in "choices."
Anyway, for regular stock buying/shorting, there are four cases:
- You buy the stock and it goes up. You win.
- You buy the stock and it goes down. You lose.
- You short the stock and it goes up. You lose.
- You short the stock and it goes down. You win.
Four cases, half are wins and half are losses. In reality it's a bit more complicated than just $50/50$ because there are many factors that influence a stock price; it's not just an equally distributed chance to increase or decrease.
John's answer is perfect, but let me analyse somewhat your problem.
For up and down you have two possibilities: $1/2$ up and $1/2$ up. You must add them: $1/2+1/2=1$.
What you mathematically mix, is that you do as if you would choose also the option by random: $1/2$ long, $1/2$ short.
Then you have four posibilities. Then the chance that you have choosen short and stock goes up, say, is: $1/2 * 1/2$, as you have written.