A, B and C are events associated with a random experiment such that P(

1.	A, B and C are events associated with a random experiment such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A ∩ B) = 0.88, P(A ∩ C) = 0.48 and P(A ∩ B ∩ C) = 0.09. If P(A ∪ B ∪ C) ≥ 0.75, then prove that P(B ∩ C) lies in the interval [0.23, 0.48].
Answer» Given, P(A) = 0.3, P(B) = 0.4, P(C) = 0.8 P(A ∩ B) = 0.08, P(A ∩ C) = 0.48, P(A ∩ B ∩ C) = 0. 09. We know– P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C) = 0.3 + 0.4 + 0.8 – 0.88 – p(B ∩ C) – 0.48 + 0.09 = 0.23 – p(B ∩ C) Since 0 ≤ P(A ∪ B ∪ C) ≤ and P(A ∪ B ∪ C) ≥ 0.75 ∴ P(B ∩ C) ≤ 0.48 and P(B ∩ C) ≥ 0.23 ∴ P(B ∩ C) lies in the interval [0.23, 0.48]

A, B and C are events associated with a random experiment such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A ∩ B) = 0.88, P(A ∩ C) = 0.48 and P(A ∩ B ∩ C) = 0.09. If P(A ∪ B ∪ C) ≥ 0.75, then prove that P(B ∩ C) lies in the interval [0.23, 0.48].

Answer»

Given,

P(A) = 0.3, P(B) = 0.4, P(C) = 0.8 P(A ∩ B) = 0.08, P(A ∩ C) = 0.48,

P(A ∩ B ∩ C) = 0. 09.

We know–

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C)

= 0.3 + 0.4 + 0.8 – 0.88 – p(B ∩ C) – 0.48 + 0.09

= 0.23 – p(B ∩ C)

Since 0 ≤ P(A ∪ B ∪ C) ≤ and P(A ∪ B ∪ C) ≥ 0.75

∴ P(B ∩ C) ≤ 0.48

and P(B ∩ C) ≥ 0.23

∴ P(B ∩ C) lies in the interval [0.23, 0.48]

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A, B and C are events associated with a random experiment such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A ∩ B) = 0.88, P(A ∩ C) = 0.48 and P(A ∩ B ∩ C) = 0.09. If P(A ∪ B ∪ C) ≥ 0.75, then prove that P(B ∩ C) lies in the interval [0.23, 0.48].

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