This article is for mathematically-literate people who want to examine their decision-making process.
Mathematically-literate means you have a good grasp of if-then logic and you can follow operations like 3(x+1)=3x+3. It also means you understand why a/2 < c/3 implies 3a < 2c (we multiplied both sides of the inequality by 6). If you like the idea of math but are not mathematically-literate, find a friend who can walk you through this article.
A basketball player is taking shots in a practice session. Early in the practice session he has a success rate of less than 80%. To make this concrete, perhaps he has taken 12 shots and made 9 of them, making his success rate 75%. Towards the end of the practice session he has a success rate of over 80%. Perhaps he has now taken a total of 60 shots and made 54 of them, making his success rate 90%.
Now we can set up the problem. Early in the practice session the basketball player has a success rate of strictly less than 80% and later in the practice session he has a success rate of strictly more than 80%. Is it guaranteed that there was a moment during the practice session when his success rate was exactly 80%?
Pause here and form an opinion before reading on. Consider how you are coming to that opinion and how you would persuade someone else that your opinion is correct.
Hopefully your opinion was in the negative, meaning you believe that it is not guaranteed that there was a moment during the practice session when his success rate was exactly 80%. I say that because I want you to be wrong. I want to change your mind. I want you to have the opportunity to examine what you may have overlooked, and how it feels to change your mind.
Most people will initially say the answer is "no." They will have the correct intuition that it's possible to skip over some percentages. Surely we can construct a sequence of shots that skips over a particular success rate like 83% for example? Absolutely.
So what did we overlook? There is something special about 80%. Maybe you had an inkling that the question was overly specific. If you did have that inkling, try to remember what it felt like because that intuition may serve you well in life.
Now let me convince you. Let's see if you are open-minded enough to change your mind. Algebra incoming!
A critical point of logic is that there is a particular shot where the player's success rate switches from under 80% to at least 80%, and this shot is a successful one (a miss would not have improved his success rate). Let's represent the under 80% success rate as x/n and the at least 80% success rate as (x+1)/(n+1). We will also represent 80% as a fraction: 4/5.
Suppose for contradiction that the player skips over the 80% success rate.
This implies x/n < 4/5 and (x+1)/(n+1) > 4/5.
Multiply the first inequality by 5n and the second inequality by 5(n+1).
This implies 5x < 4n and 5(x+1) > 4(n+1).
This implies 5x < 4n and 5x+5 > 4n+4.
This implies 5x < 4n and 5x+1 > 4n.
This is a contradiction because this says "There is a number 4n strictly between 5x and 5x+1" but 5x and 5x+1 are consecutive integers and 4n is also an integer.
We have proved that no matter what sequence of shots the player does, there must be a particular shot upon which his success rate is exactly 80%.
If you're not convinced then you either:
a) Need to work through the logic and the algebra a few more times.
b) Need to improve on changing your mind when faced with new information.