I'm just trying to visualize the costs of trading. Say I set up an account to trade something (forex, stock, even bitcoin) and I was going to let a random generator determine when I should buy or sell it. If I do this, I would assume I have an equal probability to make a profit or a loss.
Your question is what a mathematician would call an "ill-posed problem." It makes it a challenge to answer. The short answer is "no." We will have to consider three broad cases for types of assets and two time intervals.
Let us start with a very short time interval. The bid-ask spread covers the anticipated cost to the market maker of holding an asset bought in the market equal to the opportunity costs over the half-life of the holding period. A consequence of this is that you are nearly guaranteed to lose money if your time interval between trades is less than the half-life of the actual portfolio of the market maker. To use a dice analogy, imagine having to pay a fee per roll before you can gamble. You can win, but it will be biased toward losing.
Now let us go to the extreme opposite time period, which is that you will buy now and sell one minute before you die. For stocks, you would have received the dividends plus any stocks you sold from mergers. Conversely, you would have had to pay the dividends on your short sales and received a gain on every short stock that went bankrupt. Because you have to pay interest on short sales and dividends passed, you will lose money on a net basis to the market maker. Maybe you are seeing a pattern here. The phrase "market maker" will come up a lot.
Now let us look at currencies. In the long run, if the current fiat money policy regime holds, you will lose a lot of money. Deflation is not a big deal under a commodity money regime, but it is a problem under fiat money, so central banks avoid it. So your long currency holdings will depreciate. Your short would appreciate, except you have to pay interest on them at a rate greater than the rate of inflation to the market maker.
Finally, for commodities, no one will allow perpetual holding of short positions in commodities because people want them delivered. Because insider knowledge is presumed under the commodities trading laws, a random investor would be at a giant disadvantage similar to what a chess player who played randomly would face against a grand master chess player. There is a very strong information asymmetry in commodity contracts. There are people who actually do know how much cotton there is in the world, how much is planted in the ground, and what the demand will be and that knowledge is not shared with the world at large. You would be fleeced.
Can I also assume that probabilistically speaking, a trader cannot do
worst than random? Say, if I had to guess the roll of a dice, my
chance of being correct can't be less than 16.667%.
A physicist, a con man, a magician and a statistician would tell you that dice rolls and coin tosses are not random. While we teach "fair" coins and "fair" dice in introductory college classes to simplify many complex ideas, they also do not exist. If you want to see a funny version of the dice roll game, watch the 1962 Japanese movie Zatoichi. It is an action movie, but it begins with a dice game.
Consider adopting a Bayesian perspective on probability as it would be a healthier perspective based on how you are thinking about this problem. A "frequency" approach always assumes the null model is true, which is what you are doing. Had you tried this will real money, your model would have been falsified, but you still wouldn't know the true model.
Yes, you can do much worse than 1/6th of the time. Even if you are trying to be "fair," you have not accounted for the variance.
Extending that logic, then for an inexperienced trader, is it right to
say then that it's equally difficult to purposely make a loss then it
is to purposely make a profit? Because if I can purposely make a loss,
I would purposely just do the opposite of what I'm doing to make a
profit. So in the dice example, if I can somehow lower my chances of
winning below 16.6667%, it means I would simply need to bet on the
other 5 numbers to give myself a better than 83% chance of winning.
If the game were "fair," but for things like forex the rules of the game are purposefully changed by the market maker to maximize long-run profitability. Under US law, forex is not regulated by anything other than common law. As a result, the market maker can state any price, including prices far from the market, with the intent to make a system used by actors losing systems, such as to trigger margin calls. The prices quoted by forex dealers in the US move loosely with the global rates, but vary enough that only the dealer should make money systematically. A fixed strategy would promote loss.
You are assuming that only you know the odds and they would let you profit from your 83.33 percentage chance of winning.
So then, is the costs of trading from a purely probabilistic point of
view simply the transaction costs? No matter what, my chances cannot
be worse than random and if my trading system has an edge that is
greater than the percentage of the transaction that is transaction
cost, then I am probabilistically likely to make a profit?
No, the cost of trading is the opportunity cost of the money. The transaction costs are explicit costs, but you have ignored the implicit costs of foregone interest and foregone happiness using the money for other things.
You will want to be careful here in understanding probability because the distribution of returns for all of these assets lack a first moment and so there cannot be a "mean return." A modal return would be an intellectually more consistent perspective, implying you should use an "all-or-nothing" cost function to evaluate your methodology.