Probability rules and Venn diagrams

For any two events $A$ and $B$:

$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$

The overlap is subtracted because it is counted twice when you add $P(A)$ and $P(B)$.

Example. $P(A)=0,4$, $P(B)=0,35$ and $P(A\text{ and }B)=0,1$.

$P(A\text{ or }B)=0,4+0,35-0,1=0,65$

Mutually exclusive events cannot happen at the same time: $P(A\text{ and }B)=0$, so

$P(A\text{ or }B)=P(A)+P(B)$

Complementary events are mutually exclusive and together cover every outcome:

$P(\text{not }A)=1-P(A)$

Example. A learner is chosen at random. "In Grade 11" and "in Grade 12" are mutually exclusive. "Passed" and "did not pass" are complementary.

"At least one" questions are usually quickest with the complement: $P(\text{at least one})=1-P(\text{none})$.

Events are independent if one happening does not change the probability of the other:

$P(A\text{ and }B)=P(A)\times P(B)$

Use this as a test: calculate both sides. If they are equal, the events are independent.

Example. $P(A)=0,3$ and $P(B)=0,5$, with $A$ and $B$ independent.

• $P(A\text{ and }B)=0,15$
• $P(A\text{ or }B)=0,3+0,5-0,15=0,65$
• $P(\text{neither})=1-0,65=0,35$

Do not confuse independent with mutually exclusive. Mutually exclusive events with non-zero probabilities are never independent.

A Venn diagram shows overlapping events. Fill it in from the inside out:

1. Start with the intersection of all the events.
2. Fill in the regions for exactly two events.
3. Fill in each event's "only" region by subtracting.
4. The region outside all circles makes the total 1 (or the total number).

Example. Of 120 learners, 70 play sport, 50 do music and 25 do neither. Let $x$ do both:

$(70-x)+x+(50-x)+25=120\Rightarrow x=25$

So 45 play sport only, and $P(\text{sport only})=\frac{45}{120}=0,375$.

With three events, label every region and use given relationships (such as independence or mutually exclusive events) to form equations.