The Game Show Problem

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The famous Game Show Problem, also known as the Monty Hall problem, shows how easily we can misinterpret information.

This is a very important lesson for sports bettors to learn.

Failing to recognise value in the markets will mean you will lose in the long run.

The Game Show Problem

Contestants are presented with three doors.

Monty Hall Problem - 3 Doors

A new car is hidden behind one of these three doors.

Behind the other doors is a goat.

You must choose the correct door to win the car, but you don’t have any other information to help with your decision.

After choosing a door, the Game Show host opens one of the other two doors to reveal a goat.

Monty Hall Problem - Goat Revealed

You now have a choice:

Do you stay with your original decision or do you switch?

Now, the solution to this problem is quite simple but many people still struggle.

The Game Show Problem demonstrates how the average person thinks ie counter-intuitively…

… and this is also true for recreational sports bettors.

In fact, when this problem was first published in Parade Magazine, 10,000 readers actually complained that the answer was wrong – most notably professors of mathematics.

The Game Show Problem Solution

The solution:

Always switch.

Here’s why.

After you open the first door, the new car must be behind one of the two remaining closed doors.

Now, most people automatically assume there is no advantage to switching.

They think that no matter the choice, you have a 1/3 (33.33%) chance of being right.

This is incorrect and only applies before you make your first choice.

In fact, you double your chances of winning (66.6%) by switching.

Think of it like you’re choosing between:

  • Your original door – a 33.3% chance of being right; or
  • The combined probabilities of the other two doors – a 66.6% (33.3% + 33.3%) chance of being right.

The remaining doors pair together once you’ve made your first choice.

So, even when the Game Show host reveals a goat behind one one of the other doors, the chances of choosing the right door remain at 66.6%.

The graphic below helps to illustrate:

The Monty Hall Problem From A Different Viewpoint

Let’s look at this another way.

Imagine there were 100 doors to choose from instead of 3.

The Game Show host asks you to pick 1 of these 100 doors as normal.

So you pick a door.

The host then looks at the other 99 doors, finds the goats and opens 98 of them, leaving 1 unopened.

You’re then asked if you’d like to stay with your original choice or switch.

What do you do?

Do you stick with your 1/100 choice or do you go for the filtered door from the other 99?

Filtered is always better.

The host removed 98 of the 99 doors, thus improving your chances – you’re getting the best door from the 99.
So, back to the decision:

Stick with your original 1/100 choice or the best choice from 99?

In other words, you can stick with your random choice or make a filtered choice.

Why The Monty Hall Problem Is Not 50/50

This is the biggest issue that people face with this problem.

When you have two choices available and you have no information about them, then you can assume they are equally likely.

For example, if you asked non-football fans who they thought was better between Patrick Mahomes and J’Mar Smith, they wouldn’t have a clue and would choose based on random selection alone.

Then you explain that Patrick Mahomes is the reigning Super Bowl MVP, while J’Mal Smith is unproven at this level.

More Information Leads To Better Decisions

Would that change their mind?

Sure; they’ll almost certainly pick Mahomes.

Information matters.

Know The Odds

As you can see, it’s easy to assume that non-random information is actually random.

The reality is that you need to pay attention to the information in front of you and process it properly.

Sports bettors make this mistake far too often, particularly when they’re encouraged not to think of betting as a mathematical problem.

Every bet you make should be based on whether or not there’s value.

Calculate the EV and go from there.

If you don’t, in the long run, you’ll end up with the goat rather than the brand new car.

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