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Standard XII

Mathematics

Complement of an Event

If A and B ar...

Question

If A and B are two events associated with a random experiment such that
P(A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5, find P (A ∩ B).

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Solution

Given:
P(A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5
By addition theorem, we have:
P (A ∪ B) = P(A) + P (B) $-$ P (A ∩ B)
⇒ 0.5 = 0.3 + 0.4 $-$ P (A ∩ B)
⇒ 0.5 = 0.7 $-$ P (A ∩ B)
⇒ P (A ∩ B) = 0.7 $-$ 0.5
= 0.2
Hence, P (A ∩ B) = 0.2

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(i)	$\frac{1}{3}$	$\frac{1}{5}$	$\frac{1}{15}$	......
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(a) If A and B be mutualy exclusive events associated with a random experiment such that P(A) = 0.4 and P(B) = 0.5, then find :
(i) $P (A \cup B)$ (ii) $P (¯ ¯¯ ¯ A \cup ¯ ¯¯ ¯ B)$

(iii) $P (¯ ¯¯ ¯ A \cup B)$ (iv) $P (A \cup ¯ ¯¯ ¯ B)$

(b) A and B are two events such that P(A)= 0.54, P(B) = 0.69 and $P (A \cup B)$ = 0.35. Find:

(i) $P (A \cup B)$ (ii) $P (¯ ¯¯ ¯ A \cup ¯ ¯¯ ¯ B)$

(iii) $P (A \cup ¯ ¯¯ ¯ B)$ (iv) $P (B \cup ¯ ¯¯ ¯ A)$

P(A) P(B)

$P (A \cap B)$ $P (A \cup B)$

(i) $\frac{1}{3}$ $\frac{1}{5}$ $\frac{1}{15}$ ___

(ii) 0.35 ___ 0.25 0.6

(iii) 0.5 0.35 ___ 0.7

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If A and B are two events associated with a random experiment such that P(A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5, find P (A ∩ B).

Given: P(A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5 By addition theorem, we have: P (A ∪ B) = P(A) + P (B) - P (A ∩ B) ⇒ 0.5 = 0.3 + 0.4 - P (A ∩ B) ⇒ 0.5 = 0.7 - P (A ∩ B) ⇒ P (A ∩ B) = 0.7 - 0.5 = 0.2 Hence, P (A ∩ B) = 0.2

If A and B are two events associated with a random experiment such that
P(A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5, find P (A ∩ B).

Given:
P(A) = 0.3, P (B) = 0.4 and P (A ∪ B) = 0.5
By addition theorem, we have:
P (A ∪ B) = P(A) + P (B) $-$ P (A ∩ B)
⇒ 0.5 = 0.3 + 0.4 $-$ P (A ∩ B)
⇒ 0.5 = 0.7 $-$ P (A ∩ B)
⇒ P (A ∩ B) = 0.7 $-$ 0.5
= 0.2
Hence, P (A ∩ B) = 0.2