Probability
If you are familiar with probability you may skip this section.
Probability is a numerical measure of how likely an event is to occur. Imagine you have a fair coin and want to find out the probability (likelihood) that the coin lands on heads or tails in \(N\) flips: \(\frac{Heads}{Flips}\) and \(\frac{Tails}{Flips}\). You proceed to run five trials of an experiment, flipping a coin ten times more each trial and marking all the times it has landed on heads or tails.
| Flips | Heads | Tails | \(\frac{Heads}{Flips}\) | \(\frac{Tails}{Flips}\) |
|---|---|---|---|---|
| \(1*10^0\) | 1 | 0 | 1 | 0 |
| \(1*10^1\) | 3 | 7 | \(\frac{3}{10}\) | \(\frac{7}{10}\) |
| \(1*10^2\) | 52 | 48 | \(\frac{52}{100}\) | \(\frac{48}{100}\) |
| \(1*10^3\) | 487 | 513 | \(\frac{487}{1000}\) | \(\frac{513}{1000}\) |
| \(1*10^4\) | 4983 | 5017 | \(\frac{4983}{10000}\) | \(\frac{5017}{10000}\) |
With this data, you can see a pattern starting to merge. The more times we flip the coin, the closer \(\frac{Heads}{Flips}\) and \(\frac{Tails}{Flips}\) approaches \(\frac{1}{2}\). The probability that heads or tails occurs is derived by assuming that nearly infinite flips have occurred, which would give us \(\frac{1}{2}\) for both heads and tails. This is known as the Law of Large Numbers.
However, rather than using fractional notation, mathematician instead formally define it as \(P(E)\), the probability of a variable event \(E\) occurring. With \(H = Heads\) and \(T = Tails\) as the two outcomes (events) of the coin, we can then say that the \(P(H) = \frac{Heads}{Flips} = \frac{1}{2}\) and \(P(T) = \frac{Tails}{Flips} = \frac{1}{2}\).
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