IN A CLASS OF 130 STUDENTS. 85 PASS IN MATH, 60 PASS IN SCIENCE, 10 FAIL BOTH. a) how many passed in science but failed in math
b) how many failed in exactly one subject If 10 failed in both, it means that 130-10=120 passed at least one subject.
Since 85 passed math, 60 passed science, there are 145 passes from 120 students. We conclude that 145-120=25 passed both.
So (b) 120-25=95 passed (or failed) exactly one subject.
For (a), 60 passed science and 25 passed both, so 35 passed science but did not pass math.
A more mathematical treatment would be:
P'(A∪B)=1-P(A∪B)
P(A)=85/130
P(B)=60/130
P'(A∪B)=1-P(A∪B)=10/130
=>
P(A∪B)=(130-10)/130=120/130
We also know that:
P(A∪B)=P(A)+P(B)-P(A∩B)
120/130=85/130+60/130-P(A∩B)
=>
P(A∩B)=85/130+60/130-120/130=25/130
The cardinality of a set is the number of members in the set.
For example, the cardinality of A is denoted |A|, which is equal to the number of students who passed math=85.
Hence |B|=60, |A∩B|=25, |A∪B|=120.
(a)
|A'∩B|=|B|-|A∩B|=60-25=35
(b)
|A∪B|-|A∩B|=120-25=95 Set