What is Probable? (Part 2)

Further to the concept discussed in Part 1, regarding the Pick-A-Box Puzzle. There exist on the internet a number of forums debating this issue, and some of the arguments for the final decision of which box to pick, i.e. whether to switch or stick being a 50-50 decision are seductive, to the point where some people are simply unable to see that the choice remaining after a losing box has been revealed is far from even. Since our ability to assess Probability is vital if we want to avoid common pitfalls in punting, I’ll discuss this concept a bit further.

The most frequent stumbling block in this question revolves around the perception that as there are only two boxes left to choose from, after a losing box from the two non-chosen ones is revealed, that it must be a 50-50 chance as to which is the winner. The fallacies with this viewpoint are as follows:-

First, the initial choice is made from three, therefore the odds of being correct can never exceed 1/3. This is irrespective of any other information being received or not, or any actions taken by the Host. If there were 10 boxes, the odds of picking the winner with only one pick is clearly 1/10. There would be a 90% chance that the other boxes contained the winner.  If the Host revealed 8 of those boxes to be losers, would it be wise to think that your first pick had suddenly increased its Probability from 1/10 to 1/2?

In the sense that there are only two boxes left, it is correct to say that the first pick either is or is not the winner, but this is far from the same thing as a 50-50 split; the odds are still 90% that it is the other one, and if the experiment was run hundreds of times that is exactly what would transpire. While it is undeniably true that the original choice could, indeed will, sometimes win, the Probabilities dictate absolutely that 9 times out of 10 it will not. The same principle applies with three boxes; after the Host removes one of the non-chosen boxes, the remainder has the 2/3 Probability of the original two, devolving onto it alone.

We can perhaps get a clearer view by extrapolating the question into everyday life. Is there or is there not a God? The answer is a choice of two, but who could say that the chance of God existing is exactly 50%?

Or perhaps we should look at the weather. On any given day, wherever we may happen to be on the Earth’s surface, it will either rain or not rain. But who would be naive enough to argue that this was a 50-50 chance? It obviously depends upon the circumstances prevailing, and would seldom be 50%, but would almost always be weighted in favour of one side or the other.

What about the possibility of death? On any given day, we will either die or not die, but is that a 50-50 chance? We think not, or most of us would struggle to make it past our first 10 days….

Thus it is with the boxes; revealing a loser changes everything, except the original odds of the chosen box of 1/3: that can never change.

The confusion here stems from what logician’s call “Confusing a Necessary with a Sufficient Condition”. Of course, for there to exist a 50-50 chance of an outcome, there must of necessity be two possibilities, but this condition alone is not sufficient to guarantee that any choice of two options has an equal chance of success.                                                                                                     Some of us have seen the game of Crown and Anchor, or its variants. In the past, in many hometown hotels there were small Gaming rooms where the three dice bearing the symbols of the Aces of the four card suits and a crown and anchor were tossed in a locked cage. Patrons bet on any of the six symbols, and if their chosen symbol was thrown they were paid once for a single showing, twice for two showings, and the princely sum of three times if all three dice display the chosen symbol! How generous of the house to pay 3-1 for an outcome that has a Probability of 1/216! Even if one bet all six symbols every time, the Probability is only 1 in 36 that any of them will be displayed on each of the three dice. Small wonder that the game has long been illegal in most civilized parts of the world.

Now having said all that (and having had a little fun at the expense of those who give no consideration whatsoever to the concept of Probability, in case it interferes with their desire to have a bet), there still remains the consideration of things which can happen to our disadvantage.

Even though we may do everything in our power to maximize the combination of Probability and dividends to our advantage, we nevertheless have to accept the reality that we are gambling and can’t be sure of any particular outcome.

Since we are looking for value bets (the win dividend average here is about $12), we have to face the Probability, if we can achieve even a 1-in-5 success rate (which would give a POT of over 200%), that we will encounter losing streaks of 20-30 from time to time. By using a small percentage of the bank as our base bet, these losing streaks will result in our bets decreasing, while a winning streak will see our bets increase, and a decent price or two will see us recover most if not all of our lost ground.

This is why we consider it vital to include the place component as part of the strategy. Looking through the PDF of the results, one can see that even when there is a significant run of losses in the Win column, profits can still be made by including the Place bet. It’s far more satisfying to get something back from our runners when they put up a very good show, only perhaps to be pipped at the post, than it is to grumble about our ‘bad luck’.

Furthermore, we need to be prepared mentally and emotionally to endure these painful runs. Although by using a percentage of the bank, we should in theory never run out of funds (like the frog which always hops halfway to the edge of the pond, never actually reaching it), we need to be prepared that we could, if all goes horribly wrong, effectively lose our bank. By effectively, we mean that our funds shrink so much that it would probably be quicker to earn another one than win it back, with what we have remaining.

If we prepare adequately, this Probability should be negligible, but even negligible Probabilities can occur. For this reason, it is perhaps over-optimistic to hope that we can win a vast fortune from punting. It’s much more likely that we can get into a position of augmenting our normal sources of income, with the option of withdrawing funds from time to time to facilitate greater enjoyment of some aspects of life.

If we wish to analyse Probabilities sensibly, we need to cultivate rationality in as many areas of our thinking as we can, and this includes not indulging in airy flights of fancy about winning huge amounts quickly. It’s probably safe to say that if this could be done (to any lasting effect, rather than the odd Lotto-type win which people are usually unable to repeat), it would have been formulated by now, and the racing industry would have died out through lack of bookmakers.

We must be dispassionate in our analysis and reaction to results. If we’ve done our job correctly, we have to wait for the Probabilities to do theirs, and that requires patience and coolness. We are approaching our punting as a business, which means we’re operating to a plan, both with our selection process and our staking method. A business wouldn’t have much chance of survival if the proprietors constantly changed their modus operandi with every little hiccup along the way. We need a plan for whatever may happen, and this includes what to do if our bank is threatened because we may have a flaw in our reasoning. Do we go for more financing, or decide that we ought to be doing something else? The latter is a perfectly acceptable option if it’s what we have decided beforehand as part of a plan.

The important thing is that we have control over our betting, not the other way around. The Probability is very close to 1 that those who can’t control themselves when betting will ultimately fail.