Combining events
Choose the event before choosing the calculation. Marginal totals overlap; disjoint region totals do not
- Inclusive or A or B includes the overlap. Add the marginals and remove one copy of the overlap \(\displaystyle \begin{aligned}&\Pr(A\cup B)\\&\quad=\Pr(A)+\Pr(B)\\&\qquad-\Pr(A\cap B)\end{aligned}\)
- Exactly one Keep A-only and B-only. Remove both copies of the overlap from the sum of marginals \(\displaystyle \begin{aligned}&\Pr(\text{exactly one})\\&\quad=\Pr(A)+\Pr(B)\\&\qquad-2\Pr(A\cap B)\end{aligned}\)
- Neither Neither is the complement of the union \(\displaystyle \Pr(\text{neither})=1-\Pr(A\cup B)\)
- Mutually exclusive The intersection is empty, so the general addition rule reduces to adding the two event probabilities \(\displaystyle \Pr(A\cap B)=0\)
Worked example
A collection contains 32 numbered cards. Set A contains 18 cards, set B contains 15 cards and their intersection contains 7. Select one card uniformly. Find the probability of neither event
- \(\displaystyle \begin{aligned}N(A\cup B)&=18+15-7\\&=26\end{aligned}\)Subtract the overlap once
- \(\displaystyle N(A^{\prime}\cap B^{\prime})=32-26=6\)Six cards lie outside the union
- \(\displaystyle \Pr(A^{\prime}\cap B^{\prime})=\frac{6}{32}=\frac{3}{16}\)Divide by the universe count and simplify
Common mistakes
Difficulty levels
Questions about this topic
Why subtract the intersection once?+
Adding the two marginals counts each shared item twice. Subtract one copy to keep it once in the union
Why does exactly one subtract it twice?+
Exactly one excludes the intersection completely. Both copies included by the marginal sum must be removed
How can I recover the intersection?+
Add the two marginal counts and subtract the union count. The same rearrangement works for probabilities
Can the addition rule be used for disjoint events?+
Yes. The intersection term is zero, so it reduces to adding the two event probabilities
Does not A mean B?+
Only if B is exactly the complement of A in the given universe. Generally not A includes B-only and neither
Does overlap prove independence?+
No. Overlap only means the events can occur together. Independence needs a separate probability test, taught later
How can I check whether given counts are possible?+
The union must be at least as large as either set and no larger than the universe. Every recovered disjoint region must be nonnegative
Are observed proportions exact probabilities?+
Only for a stated uniform selection from that recorded finite collection. Using observed data to predict a future process gives estimates, not a guarantee
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 4: Probability