One of the weirdest cases of where common sence and probablilty disagree is the Monty Hall Paradox. The classic paradox involves a game show, a cheesy orange-tanned game show host, three boxes, a set of holiday tickets and two goats.

Imagine you have reached the final of a game show, the cheesy boot-black-haired game show host gives you a choice of three boxes. In two there are goats, in the other one there are four tickets for a dream holiday. The host knows where the ticekts are. You choose one box. The cheesy teeth-whitened game show host opens one of the other boxes and reveals to you that there is a goat in it. He now offers you the choice to stick with your current box or change to the other unopened one. Now assumeing that you would prefer a dream holiday to a goat, should you stick or should you change?

Common Sence

Common Sence would say that there there is a 50/50 chance that your box has a goat in it, and therefore a 50/50 chance that it is your dream holiday instead. Therefore it doesn’t matter if you stick or move, the odds are the same either way.

Probability Method One

The key to this is realising that since our host is not working at random, but reacting to our choice, he does not affect the initial odds.

Whatever box we choose, there will be a 1/3 chance of it containing a holiday and 2/3 chance of a goat. Put another way, there is a 2/3 chance that a different box will have the holiday.

After our sharp-suited host does his thing, the odds DO NOT change. There is still a 1/3 chance of a holiday in your box and a 2/3 chance of it not being there. Therefore the odds would say you have double the chance of getting the holiday if you switch boxes.

Probability Method Two

Let’s look at this in a different way. We will put the prizes in their boxes. For this Boxes A and C will have goats in them and Box B will have the holiday. Remember, you do not know what is in each box, but the host does. These are the events that could happen:

You chose Box A, the host opens box C. If you stick: you lose. If you swap: you win

You chose Box B, the host can open either box A or C. If you stick: you win. if you swap: you lose.

You chose Box C, the host opens box A. If you stick: you lose. If you swap: you win

Only one out of three times does sticking make you win. Two out of three times swapping makes you win. Again, the odds would say that you should swap.

Really?

Yes, Really.

This bizarre freak of mathematics has been tested and does work. To make more light of it, imagine that there are not just three boxes, but 10,000 boxes and 9,999 contain goats. If our host make a big show of opening up 9,998 of the boxes to reveal goats, would you stick or swap? You would most likely swap without thinking about its.