Why is trading in the stock market 50% probability instead of 25% probability?

Question

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?

Answer 1

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.

Answer 2

No, it's $50/50$ as you expected.

In broad strokes, if you buy, and the stock goes up, you win. If you short-sell, and the stock goes down, you win. Otherwise, you lose.

You understood the probability part, just not the how-stocks-work part.

Answer 3

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:

1. You buy the stock and it goes up. You win.
2. You buy the stock and it goes down. You lose.
3. You short the stock and it goes up. You lose.
4. 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.

Answer 4

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.