Probability

Probability

Probability is just favourable ÷ total once you count carefully — so it leans heavily on the P&C skills next door. The fast routes are the complement and the tree.

Core ideas

$P(E)=\frac{\text{favourable outcomes}}{\text{total outcomes}},\qquad 0\le P(E)\le 1.$

• Complement: $P(\text{not }E)=1-P(E)$ — the engine behind "at least one."
• Independent events (AND): $P(A\cap B)=P(A)\,P(B)$.
• Mutually exclusive (OR): $P(A\cup B)=P(A)+P(B)$.
• Conditional: $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$.

A worked example

Two fair dice are rolled. What is the probability that the sum is 7?

Total outcomes $=6\times6=36$. The favourable pairs are $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$ — six of them:

$P(\text{sum}=7)=\frac{6}{36}=\frac16.$

Seven is the most likely sum precisely because it has the most pairs — a fact worth remembering for dice questions.

Common traps

• Adding non-exclusive probabilities. If $A$ and $B$ can both happen, subtract the overlap: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
• Independent vs mutually exclusive — opposite ideas: exclusive events can't co-occur, independent ones don't influence each other.
• With vs without replacement changes the denominator on the second draw.

Checklist

• Count total and favourable with care (P&C)
• Use $1-P(\text{none})$ for "at least one"
• Multiply for independent, add for mutually exclusive
• Adjust for replacement between draws