Question

Five student $A,B,C,D$ and $E$ are competeing in a long distance race. Each students probability of winning the race is given below:
$A\to 20\%,B\to 22\%,C\to 7\%,D\to 15\%\text{and}E\to 36\%$

(v) Let $S$ be the event that $B$ will win the race​
(a). Find $P\left(S\right)$.
(b). State in words, the complementary event $S^{\prime }$.
(c). Find $P\left(S^{\prime }\right)$​

Answer 1

(a) Finding $P\left(S\right)$

$S=$ event that $B$ will win the race

$P\left(S\right)=22\%$

$=\frac{22}{100}$

$=\frac{11}{50}$

Hence, $P\left(S\right)=\frac{11}{50}$

(b) State in words, the complementary event $S^{\prime }$
$S=$ event that $B$ will win the race
Then, complementary event of $S$ will be
$S^{\prime }=$ event that $B$ will not win the race.

(c) Finding $P\left(S^{\prime }\right)$

$P\left(S\right)=\frac{11}{50}$

We know that

$P\left(S\right)+P\left(S^{\prime }\right)=1$

$\Rightarrow \frac{11}{50}+P\left(S^{\prime }\right)=1$

$\Rightarrow P\left(S^{\prime }\right)=1-\frac{11}{50}$

$\Rightarrow P\left(S^{\prime }\right)=\frac{50-11}{50}$

$\Rightarrow P\left(S^{\prime }\right)=\frac{39}{50}$

Hence , $P\left(S^{\prime }\right)=\frac{39}{50}$​​​​​​​​​​​​​​