We recently had a lecture on counter-intuitive maths. Walking into this lecture I had no idea what the term counter-intuitive maths meant. Counter-intuitive maths is something that goes against what you thought it would be or your intuition. The aim of the lecture was for us to be less confused about why somethings happen in maths that we don’t expect and how a knowledge in the fundamental concepts in mathematics can assist us in our understanding of counter-intuitive maths.

We started off the lecture by looking at the concept of coin flipping. When you flip a coin you expect that there is a 50/50 chance that you will get the outcome that you want. However if you flipped a coin 100 times would the outcome be 50 heads and 50 tails. We tested out this ourselves by flipping a coin 20 times.

I found that the coin landed on heads 8 times whilst it landed on tails 12 times. Which is a ratio of 2:3. Most people think of randomness within the context of a coin flip being the same amount of heads as tail however it is much more random than that. There are different factors that influence the outcome a coin toss such as if the coin is tossed, spun or allowed to land on the floor. All of these things influence the outcome and change the probability. For example if a coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. The video below explains how coin flips aren’t always 50/50

We went on to look at the Monty Hall problem. Monty Hall was the host of the game show Let’s Make a Deal. On the show Monty would show a contestant 3 doors. Behind two doors are goats and behind one door is a car. The contestant would pick a door. Monty would then reveal one of the doors which has a goat behind it leaving two doors left over. Monty would then ask the contestant if they would like to stick with their original decision or switch to the other door. Most people believe that because one door is taken away their 1/3 chance of picking the correct door would change to a ½ chance. This however is not true. At the beginning of the game each door has a 1/3 of being the door with a car behind it. Meaning that there is a 1/3 chance that the door you picked initially has a car behind it. When a door is taken away you still have a 1/3 chance of a car being behind the door you picked but the chance of the door you didn’t pick now has a chance of 2/3. This concept is easier to understand when it is on a larger scale. Imagine there were 100 doors and 1 of those 100 doors had a car behind it and the other 99 had goats. Say you picked door number 1 and Monty revealed 98 of the doors that had goats behind them and asked if you wanted to change your answer would you? Of course you should because the door you initially picked has a 1/100 chance of being the right door whilst the remaining door now has a 99/100 chance of being the door with a car behind it as it has absorbed all the chances of the other doors. In short if one door is taken away and you are asked if you want to switch you should switch every single time. The video below explains this way better than I ever could.

But how does counter-intuitive maths relate back to the fundamental basics of mathematics? A knowledge of basic mathematic principles is needed to fully understand counter-intuitive maths. Chance and probability play a big role in counter-intuitive maths and so without a basic knowledge of it you would never be able to understand why a coin flip is completely random or why you double your chances when you change your answer.