First slide

Theorems of probability

Question

$If A and B are two independent events such that P (A \cap B) = 2 / 15 and P (A \cap B) = 1 / 6, then P (B) is$

Moderate

A

1/5
B

1/6
C

4/5
D

5/6

Solution

$Let P (A) = x and P (B) = y . Since A and B are$ $independent events, P (A \cap B) = 2 / 15$
$\begin{array}{ll} \Rightarrow P (A) P (B) = 2 / 15 \\ \Rightarrow (1 - P (A)) P (B) = 2 / 15 \\ \Rightarrow (1 - x) y = 2 / 15 - - - - (1) \end{array}$
$\begin{array}{l} and P (A \cap B) = \frac{1}{6} \Rightarrow P (A) P (B) = \frac{1}{6} \\ \Rightarrow x (1 - y) = \frac{1}{6} \\ \Rightarrow x - x y = \frac{1}{6} - - - - (2) \end{array}$
Subtracting (1) from (2), we get
$x - y = \frac{1}{30} \Rightarrow x = \frac{1}{30} + y$
Putting this value of x in (1), we get
$\begin{array}{ll} y - y (\frac{1}{30} + y) = \frac{2}{15} \Rightarrow 30 y - y - 30 y^{2} = 2 / 5 \\ \Rightarrow 30 y^{2} - 29 y + 4 = 0 \Rightarrow (6 y - 1) (5 y - 4) = 0 \\ \Rightarrow y = 1 / 6 or y = 4 / 5 \Rightarrow P (B) = 1 / 16 or P (B) = 4 / 5 \end{array}$

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