Single Events & the Complement

Section 1 of 2

Probability from a Group

Count what you want

$P(\text{event}) = \dfrac{\text{number you want}}{\text{total number}}$.

A spinner has $4$ equal sections: two are $A$, one is $B$, one is $C$. Spin it once. Find (i) $P(A)$; (ii) $P(B)$.

(i) $P(A) = \tfrac{2}{4} = \tfrac{1}{2}$

(ii) $P(B) = \tfrac{1}{4}$

$\tfrac{1}{2},\ \tfrac{1}{4}$

A bag has $4$ blue and $3$ green marbles. One is taken out. Find (i) $P(\text{blue})$; (ii) $P(\text{green})$.

Total $= 4 + 3 = 7$

(i) $P(\text{blue}) = \tfrac{4}{7}$

(ii) $P(\text{green}) = \tfrac{3}{7}$

$\tfrac{4}{7},\ \tfrac{3}{7}$

A car park has $5$ black, $3$ red and $4$ white cars. Find (i) $P(\text{red})$; (ii) $P(\text{white})$.

Total $= 5 + 3 + 4 = 12$

(i) $P(\text{red}) = \tfrac{3}{12} = \tfrac{1}{4}$

(ii) $P(\text{white}) = \tfrac{4}{12} = \tfrac{1}{3}$

$\tfrac{1}{4},\ \tfrac{1}{3}$

Section 2 of 2

1 minus the probability

$P(\text{does not happen}) = 1 - P(\text{does happen})$. Something impossible has probability $0$.

For the bag of $4$ blue and $3$ green marbles, find (i) $P(\text{not blue})$; (ii) $P(\text{not green})$; (iii) $P(\text{pink})$.

(i) $P(\text{not blue}) = 1 - \tfrac{4}{7} = \tfrac{3}{7}$

(ii) $P(\text{not green}) = 1 - \tfrac{3}{7} = \tfrac{4}{7}$

(iii) $P(\text{pink}) = 0$ (there are none)

$\tfrac{3}{7},\ \tfrac{4}{7},\ 0$

You try

For the car park ($5$ black, $3$ red, $4$ white), find (i) $P(\text{not red})$; (ii) $P(\text{not white})$.

Use $1 - P$, or count all the others over the total.

(i) $P(\text{not red}) = \tfrac{9}{12} = \tfrac{3}{4}$

(ii) $P(\text{not white}) = \tfrac{8}{12} = \tfrac{2}{3}$

$\tfrac{3}{4},\ \tfrac{2}{3}$

You try

For the spinner (two $A$, one $B$, one $C$), find $P(\text{not } A)$.

$1 - P(A)$.

$P(\text{not } A) = 1 - \tfrac{2}{4} = \tfrac{2}{4} = \tfrac{1}{2}$

$\tfrac{1}{2}$

Probability from a group, and the complement $P(\text{not}) = 1 - P$. Next: two events — the sample space and tree diagrams.