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## Probability: An event not happening

Consider the situation of a bag with 3 gold(G) balls and 2 silver(S) balls.

What is:

• P(G)?
• P(not G)?

$P(G)=\frac{3}{5}$, 3 gold balls and 5 balls in total.

$P(\text{not }G)=\frac{2}{5}$, 2 not gold (silver) balls and 5 balls altogether.

The important point to notice here is that the number of gold balls + number of not gold balls is equal to the total number of balls.

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

#### Notation

We can write P(not A) as $P(\overline{A})$, where the over bar means not.

Examples

If the $P(A)=0.7$, what is $P(\overline{A})$?

$P(\overline{A})=1-P(A)=1-0.7=0.3$

If the $P(win)=15\%$, what is $P(\overline{win})$

$P(\overline{win})=100\%-P(A)=100\%-15\%=85\%$

If the $P(blue)=\frac{4}{15}$, what is $P(\overline{blue})$

$P(\overline{blue})=1-P(blue)=1-\frac{4}{15}=\frac{11}{15}$