Check whether the following probabilities $P (A)$ and $P (B)$ are consistently defined
(i) $P (A) = 0.5, P (B) = 0.7$ , $P (A \cap B) = 0.6$
(ii) $P (A) = 0.5, P (B) = 0.4$ , $P (A \cup B) = 0.8$

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Solution

(i) $P (A) = 0.5, P (B) = 0.7$ , $P (A \cap B) = 0.6$
It is known that if $E$ and $F$ are two events such that $E \subset F$ then $P (E) \leq P (F)$
However here $P (A \cap B) > P (A)$
Hence $P (A)$ and $P (B)$ are not consistently defined
(ii) $P (A) = 0.5, P (B) = 0.4$ , $P (A \cup B) = 0.8$
It is known that if $E$ and $F$ are two events such that $E \subset F$ then $P (E) \leq P (F)$
Here it is seen that $P (A \cup B) > P (A) a n d P (A \cup B) > P (B)$
Hence $P (A)$ and $P (B)$ are consistently defined

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Similar questions

P (A)

P (B)

P (A) = 0.5, P (B) = 0.7, P (A \cap B) = 0.6

P (A) = 0.8, P (B) = 0.5

P (B / A) = 0.4

P (A \cap B)

P (A / B)

P (A \cup B)

If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find

(i) P(A ∩ B) (ii) P(A|B) (iii) P(A ∪ B)

A, B

C

P (A) = 0.6, P (B) = 0.4, P (C) = 0.5

P (A \cup B) = 0.8, P (A \cap C) = 0.3, P (A \cup B \cup C) \geq 0.85, P (A \cap B \cap C) = 0.2

P (B \cap C) = p_{1}

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