Question

Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5, and P(B) = p. Find p if they are

(i) mutually exclusive

(ii) independent

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Answer 1

(i) If A and B are mutually exclusive, then,

$\displaystyle\longrightarrow A\cap B=\phi$

$\displaystyle\Longrightarrow P(A\cap B)=0$

We have,

$\displaystyle\small\longrightarrow P(A\cap B)=P(A)+P(B)-P(A\cup B)\quad\quad\dots(1)$

Then,

$\displaystyle\longrightarrow 0=\dfrac{1}{2}+p-\dfrac{3}{5}$

$\displaystyle\longrightarrow 0=p-\dfrac{1}{10}$

$\displaystyle\longrightarrow\underline{\underline{p=\dfrac{1}{10}}}$

(ii) If A and B are independent, then,

$\displaystyle\longrightarrow P(A\mid B)=P(A)$

$\displaystyle\longrightarrow\dfrac{P(A\cap B)}{P(B)}=P(A)$

$\displaystyle\longrightarrow P(A\cap B)=P(A)\cdot P(B)$

$\displaystyle\longrightarrow P(A\cap B)=\dfrac {p}{2}$

Then in (1),

$\displaystyle\longrightarrow\dfrac{p}{2}=\dfrac{1}{2}+p-\dfrac{3}{5}$

$\displaystyle\longrightarrow\dfrac{p}{2}=p-\dfrac{1}{10}$

$\displaystyle\longrightarrow p-\dfrac{p}{2}=\dfrac{1}{10}$

$\displaystyle\longrightarrow\dfrac{p}{2}=\dfrac{1}{10}$

$\displaystyle\longrightarrow\underline{\underline{p=\dfrac{1}{5}}}$

Answer 2

Given :-

The events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5, and P(B) = p.

To Find :-

Find p if they are

(i) mutually exclusive

(ii) independent

Solution :-

For mutually exclusive

$\sf \dfrac{3}{5} = \dfrac{1}{2} + p-0$

$\sf \dfrac{3}{5} = \dfrac{1}{2} + p$

$\sf p =\dfrac{3}{5} - \dfrac{1}2$

$\sf p = \dfrac{6 - 5}{10}$

$\sf p = \dfrac{1}{10}$

For independent

According to the question

$\sf P(A) \times P(B)$

$\sf\dfrac{1}{2}\times p$

$\sf \dfrac{p}{2}$

Now

$\sf \dfrac{3}{5} -\dfrac{1}{2}=p-\dfrac{p}{2}$

$\sf\dfrac{6-5}{10}=\dfrac{2p-p}{2}$

$\sf\dfrac{1}{10}=\dfrac{p}{2}$

$\sf 10p=2$

$\sf\dfrac{2}{10}=\dfrac{1}{5}$