1.3.1 Independence

This doesn’t mean the same thing as being able to roam around late at night without your parents breathing down your neck. It’s more like with the missile example, or the coin tossing one. If you toss a coin once, whether or not it lands heads or tails, this doesn’t influence the outcome of any other coin tosses. So you can say that one toss is independent of all the other tosses.

This is the way it works. If the probability of one toss coming up heads (H) is $\frac{1}{2}$, then the probability of two tosses coming up heads is

$$P(H)P(H)=\frac{1}{2}\frac{1}{2}=\frac{1}{4}.$$

(1.1)

In more abstract notation, suppose you have two events, A and B. The probability of A and B we’ll denote AB. Then if A and B are independent, the probability of A and B is denoted $P(AB)$. So we can say

$$P(AB)=P(A)P(B)forindependenteventsAandB.$$

(1.2)

There is a more general way of seeing this, a theorem which is used a lot when doing probability, ”Bayes’s Theorem”. But before we do that, we want to take a detour and think about probability in terms of Venn diagrams