Answer:
The probability is 0.68
Step-by-step explanation:
We are going to use conditional probability to solve this problem.
We define the following events :
A : '' Alex replaced the starter''
B : ''Alex replaced the battery''
M : ''The engine works''
NM : ''The engine doesn't work''
Given two events A and B, we define the conditional probability:
[tex]P(A/B) =\frac{P(A,B)}{P(B)} \\P(B)> 0[/tex]
[tex]P(B/A)=\frac{P(B,A)}{P(A)} \\P(A)> 0[/tex]
Where P(A,B) = P(B,A) = P(A∩B) = P(B∩A)
In our problem :
[tex]P(M/B)=0.7\\P(M/A)=0.2\\P(B)=0.85\\[/tex]
Alex replaced the battery or either the starter ⇒
[tex]P(A)=1-P(B)=1-0.85=0.15\\P(A)=0.15[/tex]
We need to find [tex]P(B/NM)=\frac{P(B,NM)}{P(NM)}[/tex]
We write :
[tex]P(M/B)=\frac{P(M,B)}{P(B)} \\0.7=\frac{P(M,B)}{0.85}\\ P(M,B)=0.595[/tex]
[tex]P(M/A)=\frac{P(M,A)}{P(A)} \\0.2=\frac{P(M,A)}{0.15} \\P(M,A)=0.03[/tex]
P(M) = [P(M∩B) ∪ P(M∩A)]
P(M) = P(M∩B) + P(M∩A) - P[(M∩B)∩(M∩A)]
But P[(M∩B)∩(M∩A)] = 0 because he replaced the battery or either the starter
P(M) = P(M∩B) + P(M∩A)
P(M)=0.595+0.03
P(M)=0.625
P(NM)= 1-P(M)=1-0.625=0.375
P(NM)= 0.375
P(NM/B) = 1-P(M/B)=1-0.7=0.3
P(NM/B) = 0.3
[tex]P(NM/B)=0.3=\frac{P(NM,B)}{P(B)}=\frac{P(NM,B)}{0.85} \\P(NM,B)=0.255[/tex]
[tex]P(B/NM)=\frac{P(NM,B)}{P(NM)} =\frac{0.255}{0.375} =0.68[/tex]
