examples on independent

Example:- Probability that Ali passes a math exam is 4/5 and that he passes a chemistry exam is 5/6 . if the probability that he passes both exams is ¾ ,find the probability that he will pass at least one exam.

Sol:- let M = Ali pass math exam,
C = Ali pass Chemistry exam
P(Ali pass at least one exam)=p(M?C)
=P(M)+P(C)-P(M?C)
=4/5+5/6-3/4=53/60.
Example:- If the probability of success in one of the tests is equivalent to 0,6and the probability of success of Mohammed and Ahmed together in this test equal to 0.1. Find the probability of success Mohammed and failure of Ahmed.
A= {success of Mohammed}
B= {success of Ahmed}
A?B= {Success of Mohammed and Ahmed}
B^c= {failure of Ahmed}
A?B^c= {success of Mohammed and failure of Ahmed}
P (A) =0.6, P (A?B) =0.1
P (A?B^c) =P (A)-P (A?B)=0.6-0.1=0.5

Example:- same as in previous example, we use the definition of events as follows:

A= {success of Mohammed}, P (A) =0.25
B= {success of Ahmed}
B^c= {failure of Ahmed}, P (B^c)=0.3
A?B= {Success of Mohammed and Ahmed}
P (A?B) =0.1
Find Probability of success at least one of them.
A?B= {Success at least one of them}
= {Success of Mohammed or success of Ahmed}
P(A?B)= P(A)+P(B)-P(A?B) =0.25+0.7-0.1 =0.85
1-P (B^c) =1-0.3=0.7