Let S = {a1, a2, a3, a4}, and let P be a probability function on S.
Find P(a1) and P(a2) if P(a3) = P(a4) = 1/4 and P(a1) = 2P(a2).
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Answer:
P(a
1
)=0.3,P(a
2
)=0.1,P(a
3
)=0.4,P(a
4
)=0.2
P(A)=P(a
1
)+P(a
2
)=0.3+0.1=0.4
P(B)=P(a
3
)+P(a
4
)=0.4+0.2=0.6
P(A∪B)=1
As we know that,
P(A∪B)=P(A)+P(B)−P(A∩B)
⟹1=0.4+0.6−P(A∩B)⟹P(A∩B)=0
P(A
′
∩B
′
)=1−P(A∩B)=1−0=1
Hence P(A
′
∩B
′
)=1
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