What is the probability?

This post is about what the probability of something happening is.

For this, let's assume that we go down the road and measure the height of every adult we meet. Let's say that out of the first 30 people, 10 had a height between 160 and 170 cm. This means that 10 out of 30 or 1 out of 3 people have a height between 160 and 170 cm. We call this the relative frequency of the height 160 - 170 cm. But this may not be representative of the whole population. It may have been that the first 30 people were football club members attending a party, meaning they were of similar height. So, we keep walking, measuring the height of more and more people. After we have collected the height of 5 million people, we find that 2 million out of the 5 million have a height between 160 and 170 cm. We can now say that the relative frequency of the height 160-170 cm is 2 out of 5, which equals 0.4. Now, because we have a very big sample, we won't call this the relative frequency any more. We will call it probability. So, the probability of finding a person whose height is between 160 and 170 cm is 0.4. In other words, as we walk down the road, we have 0.4 probability that the next adult person we meet is as tall as that.

So, the probability of something happening is what its relative frequency tends to be as the sample size tends to be larger and larger. In mathematics, we call this a limit. The limit of the relative frequency as the sample size approaches infinity.

This is an example. We all know that the probability of getting number 6 after throwing a die with six sides is 1 out of 6. But we have all been in situations where we threw the die more than six times and we did not get any sixes. So, why do we say the probability is 1 out of 6? Because if we throw it thousands of times, the relative frequency will be approximately 1 out of 6. The more times we throw it, the more the relative frequency will tend to be close to 1 out of 6. If we could throw it infinite times, the relative frequency would be precisely 1 out of 6.