Probability rules

Use the complement rule and the addition rule correctly, and recognise when events are mutually exclusive.

30 minFoundation8 graded exercisesMembers

An online shop knows that 30% of its customers use the mobile app and 45% subscribe to the newsletter. What share can it reach through at least one of the two channels? Simply adding 30% and 45% double-counts the customers who do both — and the same mistake, in an insurance or risk report, can make a risk look far bigger than it is.

Every probability rule follows from three basic facts about events in a sample space $S$ :

$0 \le P(A) \le 1$ for every event $A$ .
$P(S) = 1$ : some outcome must occur.
If $A$ and $B$ cannot happen together, then $P(A \text{ or } B) = P(A) + P(B)$ .

Two pieces of notation: $A \cap B$ (“ $A$ and $B$ ”) is the event that both happen; $A \cup B$ (“ $A$ or $B$ ”) is the event that at least one happens. In probability, or is always inclusive: “or both” is included.

Included with a Formula Quest membership

It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Use the complement rule, especially for “at least one” questions
Apply the general addition rule and explain why the overlap is subtracted
Recognise mutually exclusive events and when probabilities may simply be added
Check probability rules against counts in a real dataset

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