Probability · Ordinary Level
P(A or B) & Mutually Exclusive
Ordinary Level · Class 9 · Tap NEXT to begin
Section 1 of 2
The ‘Or’ Rule
P(A or B)
$P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$ — add the two, then take away the overlap (so it isn’t counted twice).
Worked example — from a Venn diagram
A Venn diagram has $P$ only $1$, both $3$, $Q$ only $2$, outside $5$. Find $P(P \text{ or } Q)$.
Total $= 1+3+2+5 = 11$
$P \text{ or } Q = 1+3+2 = \tfrac{6}{11}$
$\tfrac{6}{11}$
Worked example — with the formula
A Venn diagram has $A$ only $3$, both $2$, $B$ only $6$, outside $5$. Find $P(A \text{ or } B)$ two ways.
Total $= 3+2+6+5 = 16$; by counting: $3+2+6 = \tfrac{11}{16}$
By formula: $P(A) + P(B) - P(A\cap B) = \tfrac{5}{16} + \tfrac{8}{16} - \tfrac{2}{16} = \tfrac{11}{16}$
$\tfrac{11}{16}$
Worked example — a card
A card is drawn from $52$. Find $P(\text{a heart or a Queen})$. ($13$ hearts, $4$ Queens, $1$ Queen of hearts.)
$P(H) + P(Q) - P(\text{Queen of hearts}) = \tfrac{13}{52} + \tfrac{4}{52} - \tfrac{1}{52}$
$= \tfrac{16}{52} = \tfrac{4}{13}$
$\tfrac{16}{52} = \tfrac{4}{13}$
Section 2 of 2
Mutually Exclusive
Nothing in common
Two events are mutually exclusive if they have nothing in the overlap — the intersection is empty. Then $P(A \text{ or } B) = P(A) + P(B)$.
You try
A Venn diagram has $A = 5$ and $B = 3$ with no overlap (total $8$). Find (i) $P(A)$; (ii) $P(B)$; (iii) are $A$ and $B$ mutually exclusive?
No overlap means nothing in the middle.
(i) $P(A) = \tfrac{5}{8}$
(ii) $P(B) = \tfrac{3}{8}$
(iii) Yes — there is nothing in the middle
$\tfrac{5}{8},\ \tfrac{3}{8},\ $ yes
That’s Class 9.
The ‘or’ rule $P(A)+P(B)-P(A\cap B)$, and mutually exclusive events (empty overlap). Class 10: expected value.