Are the Odds Against You? The Monty Hall ‘Problem’

Lets Make a Deal

In the early 1960’s, a television show called ‘Let’s Make a Deal,’ first aired. It became very popular and ran for decades. The show was first hosted by Monty Hall. In time, the ‘riddle’ presented in the game would become known as the Monty Hall ‘problem’, named after the show’s host.

In the game, a contestant is presented with three doors, behind one of which is a prize (often a car) and behind the other two are goats. The contestant is asked to choose one of the doors, that initially remains unopened. The game-show host, who knows what is behind each door, opens one of the other two doors to reveal a goat.

The host then gives the contestant the option to switch their choice to the other unopened door, or to stick with their original choice.

The Monty Hall problem appeared in the realm of mathematics in 1990. A letter found its way to Marilyn vos Savant, a columnist for an American magazine. At that time, she held the Guinness World Record for the highest recorded IQ of any living person.

The Monty Hall ‘Problem’

The letter went something like this:

Imagine that you are participating in a game show, and you are presented with three doors. One of the doors conceals a car, while the other two doors hide goats. You select one door, let’s say Door No. 1, and the host, who is aware of what lies behind each door, opens another door, Door No. 3, to reveal a goat. The host then poses the question, “Would you like to switch to Door No. 2?”

Would it be beneficial for you to change your initial choice?

Most people who encounter this problem, intuitively assume that it doesn’t matter whether they switch doors or not, as both doors have an equal probability of containing the car/prize. In one study, only 12% of participants opted to switch doors. Marilyn vos Savant concluded, however, that the contestant should always switch doors to maximise the chances of winning.

She published her answer and explanation in Parade Magazine. Over the following weeks, she received literally thousands of letters from readers, including many maths professors and educated individuals with PhDs, complaining that she had got it wrong. She was proven to be correct.

Endowment Effect

One explanation for why individuals tend to stick with their initial choice is rooted in behavioural economics. The endowment effect is a phenomenon whereby people attribute greater value to things they already own or have chosen. They prefer the current state of play, which is known as status quo bias, which leads them to avoid changing their original choice and choosing a different option.

Research has also shown that individuals prefer errors of omission over errors of commission. This means that they would rather not take action and lose, than take action and lose.

So, the Monty Hall game became a famous conundrum, that is often used to illustrate counterintuitive results in probability theory.

The Odds

• The probability of winning if you stick with your original choice/door is 1/3 or 33.33%.
  In fractional odds, this equates to 2/1 or decimal odds of 3.00.
  
• The probability of winning if you switch doors is 2/3 or 66.67%.
  In fractional odds, this equates to 1/2 or decimal odds of 1.50.

The optimal strategy is to always switch doors. You will win the prize twice as often, than if you always stick with your original choice/door.

Conditional Probability

From a probability perspective, the reason why switching is the optimal strategy can be understood through conditional probability. When the contestant first chooses a door, there is a 33.33% probability that they have chosen the ‘prize door’, and a 66.67% probability that they have chosen a door with a goat behind it.

When the host opens one of the other doors to reveal a goat, this does not change the probability that the contestant has chosen the correct door. However, it does change the probability that the other unopened door contains the prize.

Specifically, if the contestant’s original choice was a door with a goat behind it (which occurs 2/3 of the time) then the other unopened door must contain the prize, since the host is required to open a door with a goat behind it. Therefore, if the contestant switches to the other unopened door, they will win the prize 2/3 of the time, while if they stick with their original choice, they will win only 1/3 of the time.

The Monty Hall Problem and Betting

The Monty Hall problem reveals how easily people succumb to treating non-random information as if it were random. The general public tend to have a weak understanding of probability, so contestants fail to realise when they are in a statistically strong or weak situation. When faced with a dilemma like the Monty Hall problem, they will usually act on false instincts about their probability of winning.

Bettors also make these same mistakes, frequently acting against their best interests. This is especially true when their judgement is clouded by clever marketing campaigns. Often these are seen in the windows of bookmakers, along with glamourous images of a betting lifestyle. Instead, punters should be concentrating on the maths.

Punters should understand that the odds on offer need to be better than the statistical probability of that event occurring. This is the key to profit.

Conclusion

The Monty Hall ‘problem’ shows us how we often show a basic inability to accurately weigh up the chances of success. If we are given the task of selecting one good outcome, over two bad outcomes, more often than not we will get it wrong.

The Monty Hall problem can be looked at from both a betting and a probability perspective, to show that the optimal strategy is to always switch doors. This action is counterintuitive to many people, who might assume that the probability of winning is equally split between the two unopened doors, after one of them is revealed to contain a goat. However, this assumption is incorrect, and by understanding the underlying probability theory, we can see why switching is the better strategy.