TheIBAxolotl
Combining events
For two events \(A\) and \(B\):
- \(A \cap B\): both happen (intersection).
- \(A \cup B\): at least one happens (union).
Adding \(P(A)\) and \(P(B)\) counts the overlap twice, so
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Mutually exclusive events cannot happen together: \(P(A \cap B) = 0\), so \(P(A \cup B) = P(A) + P(B)\).
Conditional probability
"Given that \(B\) has happened" shrinks the sample space to \(B\):
\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]
In a Venn diagram or table, this is simply the count in the overlap divided by the count in the given region.
Independence
\(A\) and \(B\) are independent if one happening does not change the chance of the other: \(P(A \mid B) = P(A)\), which is equivalent to
\[P(A \cap B) = P(A)\,P(B)\]
To test independence, compute both sides and compare. Do not confuse independent with mutually exclusive: two events with non-zero probabilities that are mutually exclusive are never independent, because one happening makes the other impossible.
Worked example 1: Venn diagram
In a group of 80 students, 45 play football, 30 play basketball and 12 play both.
- Football only: 33; basketball only: 18; neither: \(80 - (33 + 12 + 18) = 17\).
- \(P(\text{basketball} \mid \text{football}) = \frac{12}{45} = \frac{4}{15}\).
- Independence check: \(P(F)P(B) = \frac{45}{80} \times \frac{30}{80} \approx 0.211\), but \(P(F \cap B) = \frac{12}{80} = 0.15\). These differ, so the events are not independent.
Worked example 2: tree diagram
A factory finds that 3% of phones are faulty. A test flags 95% of faulty phones and wrongly flags 2% of good ones.
- Multiply along branches: \(P(\text{faulty and flagged}) = 0.03 \times 0.95 = 0.0285\); \(P(\text{good and flagged}) = 0.97 \times 0.02 = 0.0194\).
- Add the branches you need: \(P(\text{flagged}) = 0.0479\).
- Reverse the condition: \(P(\text{faulty} \mid \text{flagged}) = \dfrac{0.0285}{0.0479} \approx 0.595\).
So only about 60% of flagged phones are actually faulty, a surprising and very examinable result.
Choosing a diagram
- Venn diagram: two or three overlapping categories with counts or probabilities.
- Tree diagram: events in stages, especially when later probabilities depend on earlier ones (drawing without replacement).
- Sample-space table: two independent experiments with equally likely outcomes, like two dice.
Common mistakes
- Forgetting to subtract the overlap when finding \(P(A \cup B)\).
- Dividing by the wrong total in conditional probability (use the "given" group).
- Not changing the numbers on the second branch for "without replacement".
- Saying events are independent because they are mutually exclusive.
Exam tips
- Fill in a Venn diagram from the centre outwards.
- On tree diagrams, check each set of branches adds to 1.
- For "show that ... are not independent", write both \(P(A)P(B)\) and \(P(A \cap B)\) as numbers and state they differ.