The Monty Hall Teaser


Most people tend to be positive, certainly when it comes to betting or gambling. After all, what is the point of getting involved if you don’t think you can win!
If you offer the casual bettor in a losing position, half of his losses back, or the alternative of betting again to clear his losses but at the risk of increasing their losing position, they will most often pick the latter option.
(For example: Bettor A is down $100. You offer him half his losses ($50) back to stop betting, so that his losing position is only $50, or the opportunity to risk another $100 to clear his current $100 loss, then it is the latter they will most often pick, regardless of whether or not it is a good ‘value’ bet).
I nickname it ‘thrill theory’ (although there are  more sophisticated terms!) simply the way in which we tend to seek out another ‘high’ and ‘fear’ losses – even if trying to eliminate losses might not be the best decision in terms of the situation or mathematical probability.
The ‘Monty Hall problem’ (as it has become known) is a great example of how, when presented with the task of choosing one favourable outcome against two unfavourable outcomes, we find it almost impossible to correctly address the likelihood of success.


Car, Goats, Doors

The Monty Hall problem is a kind of probability puzzle, so named after game show host, Monty Hall, who featured on ‘Let’s Make a Deal’ on TV back in the Sixties and Seventies.
The ‘teaser’ goes like this:
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car and behind the other two doors are goats. You pick a door, say No. 1 and the game host (who knows what’s behind the doors) opens another door, say No. 3, which reveals a goat. He then asks if you want to switch doors and pick door No.2. Is it to your advantage to switch your choice?

  1. The host must always open a door that was not picked by the contestant.
  2. The host must always open a door to reveal a goat and not the car.
  3. The host must always offer the chance to switch between the originally chosen unopened door and the remaining closed door.

Most people see no advantage in switching doors. They see it as a straight 50/50 shot as to whether they should ‘stick’ on their chosen door or switch to the remaining door. Yet they would be wrong. The probability that the car lies behind the remaining door is 66.6%. So you should always switch doors.
When the question was posed in Parade magazine, ten thousand readers wrote in to complain that the published answer was wrong, including one thousand people with PhD’s and several maths professors.
Perhaps an easier way of understanding the problem is by imagining that you’re picking between your original door (33.3% probability) and the combined probabilities of the other two doors (33.3% + 33.3%). This is because once you choose your door, the other two doors  are then paired together, so there is a 66.6% chance that the car is behind one of the remaining two doors. When one option  (goat / door) is then removed by the host, there is still a 66.6% chance that the car lies behind the remaining door.
monty hall
This is an extremely counter-intuitive problem. Which is why most people get it wrong!
Similarly the more recent UK TV show “Deal or No Deal” exploits the general public poor understanding of probability, as does the mass of advertising in bookmakers’ window on the average UK high street, which is basically designed to entice the general public into a ‘bad bet’.
If you are thinking of taking your betting seriously, then you need understand value, making sure that you have a statistical edge, that probability is on your side. Try to fully understand the bets  that you are making.
Good luck!

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