Question

Sum of the probabilities of an event and its complement is .

Answer 1

$\underset{–}{\text{Complement Rule of Probability}:}$
The Complement Rule states that the sum of the probabilities of an event and its complement must equal $1.$

For example, If we roll a fair die, then the sum of probabilities of an event "getting a $3$" and it's completement event "not getting $3$" is equal to $1.$

$\underset{–}{\text{Verification}:}$
If a fair die is rolled,

$\text{Sample Space}=\{1,2,3,4,5,6\}$
$\therefore$ Total number of possible outcomes $=6$

Favorable outcome of an event "getting $3$" $=\left\{3\right\}$

Number of favorable outcome of an event "getting $3$" $=1$

$\therefore \text{P(getting 3)}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

$\Rightarrow \text{P(getting 3)}=\frac{1}{6}$

Favorable outcome of an event "not getting $3$" $=\{1,2,4,5,6\}$

Number of favorable outcome of an event "not getting $3$" $=5$

$\therefore \text{P(not getting 3)}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

$\Rightarrow \text{P(not getting 3)}=\frac{5}{6}$

Now, $\text{P(getting 3)}+\text{P(not getting 3)}$

$=\frac{1}{6}+\frac{5}{6}$

$=\frac{1+5}{6}=\frac{6}{6}=1$

Hence, proved, sum of the probabilities of an event and its complement is equal to $1.$