Question

Alex can walk to school, cycle to school or go by bus. The probability that
he walks or cycles to school is
3 /4.
(a) If the probability that he cycles is 1/4
, what is the probability that he
walks?
(b) What is the probability that he goes by bus?
(c) What is the probability that he cycles or goes by bus?

Answer 1

Dear Student,

$$\begin{gathered} SinceAlexcanwalktoschool,takebusorcycle. \\ LetP\left(A\right)beprobabilitythathewalktoschool \\ LetP\left(B\right)beprobabilitythathetakebus \\ LetP\left(C\right)beprobabilitythatheusecycle \\ NowP\left(A\right)+P\left(B\right)+P\left(C\right)=1 \\ Nowgiventhatprobabilityhewalksorcycletoschool=\frac{3}{4} \\ P\left(A\right)+P\left(C\right)=\frac{3}{4} \\ Alsotheprobabilitythathecycles=P\left(C\right)=\frac{1}{4} \\ puttingthisinP\left(A\right)+P\left(C\right)=\frac{3}{4},weget \\ P\left(A\right)+\frac{1}{4}=\frac{3}{4} \\ \Rightarrow P\left(A\right)=\frac{3}{4}-\frac{1}{4} \\ \Rightarrow P\left(A\right)=\frac{2}{4} \\ \Rightarrow \boxed{P\left(A\right)=\frac{1}{2}} \\ b)asweknowthatP\left(A\right)+P\left(B\right)+P\left(C\right)=1 \\ \Rightarrow \frac{1}{2}+P\left(B\right)+\frac{1}{4}=1 \\ \Rightarrow P\left(B\right)=1-\frac{1}{2}-\frac{1}{4} \\ \Rightarrow P\left(B\right)=1-\left(\frac{1}{2}+\frac{1}{4}\right) \\ \Rightarrow P\left(B\right)=1-\frac{3}{4} \\ \Rightarrow \boxed{P\left(B\right)=\frac{1}{4}} \\ HenceprobabilitythathegoesbybusP\left(B\right)=\frac{1}{4} \\ c)ProbabilitythathecycleorgoesbyBus=P\left(B\right)+P\left(C\right) \\ \Rightarrow P\left(B\right)+P\left(C\right)=\frac{1}{4}+\frac{1}{4} \\ \Rightarrow P\left(B\right)+P\left(C\right)=\frac{2}{4} \\ \Rightarrow P\left(B\right)+P\left(C\right)=\frac{1}{2} \end{gathered}$$
Regards,