Monty Hall Problem and the Punter
It might seem strange that a US television show has anything to teach a modern sports bettor, but the mathematics of the Monty Hall Problem exposes a fundamental psychological aspect of betting. In fact, the Problem is a classic example of a decision-making exercise in which our natural inclinations run contrary to the logical choice, underlying how difficult it often is for us to correctly way up chance.
This has obvious and important implications for the punter because the key to success with all betting is in correctly estimating probabilities or chances. Without this ability, it is impossible for any punter to judge when a sportsbook has got their odds wrong, and so impossible to make a long-term profit.
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The Monty Hall Problem Explained
The Problem is taken from a popular US television show of the 1960s and 1970s. The show was called Let’s Make A Deal and was hosted by a presenter named Monty Hall. Contestants on the show were presented with three doors. Behind one of the doors was a prize, but behind the other two doors were goats. No other information was given to the contestants.
First, contestants were asked to pick a door. The door remained closed, and one of the other doors was opened to reveal a goat. At this point, the contestant was presented with a dilemma. Did they stick with their original choice or change to the other door? What would you choose?
The Danger of Intuition
This apparently simple challenge exposed what appears to be a fundamental weakness in human psychology. The intuitive decision is to stick with your choice, on the grounds that choosing the other door offers no advantage, and this was the option that most people seemed to choose.
In fact, when the correct answer to what had become known as the Monty Hall Problem was published in a magazine, thousands of readers complained that the answer was incorrect. The Monty Hall Problem has even been known to fool math professors.
Monty Hall: The Solution
The correct answer to the Monty Hall Problem is to always choose the other door. This seems counter-intuitive, and the majority of contestants preferred to stick with the door that they had already chosen, on the grounds that the odds of finding the prize in both cases was roughly 2/1 or 0.333.
But this misunderstands a fundamental aspect of assessing probability because while the overall odds remain the same after one door has been opened, the odds associated with the individual doors change. It is true that, when the contestant makes their first choice, the odds of finding the prize were 33.3 percent, whichever door was picked. But once one door has been opened, the odds of the car being behind the unselected door effectively double to 66.6 percent.
Monty Hall Math
The maths behind this are actually straightforward if you look at the problem logically. When the contestant chose the first door, the odds that the prize was not behind that door were effectively 66.6 percent, made up of 33.3 percent for each of the two closed doors.
When one of the other doors is opened to reveal a goat, the probability that the prize is not behind the chosen door is still 66.6 percent, but now that probability is entirely allocated to the unchosen door, making it logical for the contestant to switch their choice to that door.
Monty Hall Psychology
The Monty Hall Problem shows how easy it is for any of us to make the mistake of interpreting non-random information as if it were random. In fact, there is a more up to date version of the Monty Hall Problem in the UK show Deal or No Deal. In this game, there were 22 identical unopened boxes, each containing a different cash prize and each allocated to one contestant.
When a contestant was chosen to take part in the game, they brought their box with them and then had to choose which of the other boxes to open. At several points during the game, they were given the chance to accept a cash offer from a mysterious ‘dealer’ in exchange for giving up their box.
Anyone who watched the show will have noticed how rarely the contestants involved appeared to be weighing up their winning chances from a logical or statistical point of view, and more importantly, how frequently their decisions were motivated by intuitive or emotional sentiments.
The Lesson of Monty Hall
The Monty Hall Problem is not directly applicable as a strategy to sports betting. But it has a much more significant implication. While it won’t tell you how to weigh up a game between Barcelona and Manchester United, it can serve to remind you of the importance of taking an objective approach.
The human brain is capable of complex and sophisticated calculations, weighing up situations and reaching rapid, effective judgements, as well as being able to come up with creative and ingenious solutions to various problems. But the drawback is that we are also susceptible to emotion, to overly positive or negative attitudes, and to the whole range of subjective influences that can cloud our judgement and lead us to make reckless, ill-founded, and incorrect judgements.
The true lesson of the Monty Hall Problem is, therefore, that objectivity is the most important element of the sports betting strategies. The ability to weigh up your bet objectively is as necessary as knowing the sport you are betting on and getting the right staking plan, and by learning the lessons of a 1970s US television show, you will become a more logical punter and boost your long-term profit.