If P ( B ) = 3 4 , P ( A B C ) = 1 3 and P ( A B C ) = 1 3 , then P ( B C ) is [2003]

Question

If $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3},$ then $P (B \cap C)$ is [2003]

Smart Achievers · Accepted Answer

(1)

$Given : P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$

$P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$

$From above venn diagram, we see$

$B \cap C = B - (A \cap B \cap \bar{C}) - (\bar{A} \cap B \cap \bar{C})$

$\Rightarrow P (B \cap C) = P (B) - P (A \cap B \cap \bar{C}) - P (\bar{A} \cap B \cap \bar{C})$

$\Rightarrow P (B \cap C) = \frac{3}{4} - \frac{1}{3} - \frac{1}{3}$ $= \frac{9 - 4 - 4}{12} = \frac{1}{12}$

Solution

Q.
If $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3},$ then $P (B \cap C)$ is [2003]

1 $\frac{1}{12}$

2 $\frac{1}{6}$

3 $\frac{1}{15}$

4 $\frac{1}{9}$

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Solution

Q. If P(B)=34, P(A∩B∩C¯)=13 and P(A¯∩B∩C¯)=13, then P(B∩C) is [2003]

1 112

2 16

3 115

4 19