UPSC NDA I 2024 Mathematics Paper

We know that P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)
Substituting the given values, we get 0.85≤0.6+0.4+0.5−P(A∩B)−0.3−P(B∩C)+0.2
Simplifying the inequality, we get P(A∩B)+P(B∩C)≤0.55
We also know that P(A∪B)=P(A)+P(B)−P(A∩B)
Substituting the given values, we get 0.8=0.6+0.4−P(A∩B)
Solving for P(A∩B), we get P(A∩B)=0.2
Substituting this value into the inequality obtained earlier, we get 0.2+P(B∩C)≤0.55
Solving for P(B∩C), we get P(B∩C)≤0.35
Therefore, the maximum value of P(B∩C) is 0.35 .