Independent Events

Two events $A$ and $B$ are called independent if the occurrence of one has absolutely no effect on the probability of the other — knowing that $B$ has occurred gives no new information about $A$, so $P(A|B)=P(A)$ and $P(B|A)=P(B)$.

Substituting this into the Multiplication Theorem (Section 4) gives the defining test for independence:

This is the single most-used shortcut in this chapter: whenever a problem says two experiments happen 'simultaneously' or 'independently' (e.g. tossing a coin and rolling a die together, or two draws WITH replacement from a bag), the joint probability is simply the plain product $P(A)\times P(B)$ — no conditional term is needed at all.

Independence is not the same as mutual exclusiveness — in fact, if $P(A)>0$ and $P(B)>0$, two mutually exclusive events can NEVER be independent, because $A\cap B=\varnothing$ means $P(A\cap B)=0$, which can only equal $P(A)\times P(B)$ if one of the two probabilities is itself zero. Students preparing for the Gujarat Std-12 Statistics examination frequently confuse these two ideas — mutually exclusive events are, in fact, the most strongly DEPENDENT pair possible, since the occurrence of one makes the other's occurrence impossible. …