Question

solve with solution..​

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\:P(A) = \dfrac{1}{4}$

$\rm :\longmapsto\:P(A|B) = \dfrac{1}{2}$

$\rm :\longmapsto\:P(B|A) = \dfrac{2}{3}$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\:P(B)$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:P(A) = \dfrac{1}{4}$

and

$\rm :\longmapsto\:P(B|A) = \dfrac{2}{3}$

We know,

$\rm :\longmapsto\:P(B|A) = \dfrac{P(A\cap B)}{P(A)}$

So,

$\rm :\longmapsto\:P(A\cap B) = P(A) \times P(B|A)$

On substituting the values, we get

$\rm :\longmapsto\:P(A\cap B) = \dfrac{1}{4} \times \dfrac{2}{3}$

$\bf :\longmapsto\:P(A\cap B) = \dfrac{1}{6}$

Now, Further given that,

$\rm :\longmapsto\:P(A|B) = \dfrac{1}{2}$

We know that

$\rm :\longmapsto\:P(A|B) = \dfrac{P(A\cap B)}{P(B)}$

On substituting the values, we get

$\rm :\longmapsto\:\dfrac{1}{2} = \dfrac{\dfrac{1}{6} }{ \: \: P(B) \: \: }$

$\rm :\longmapsto\:\dfrac{1}{2} = \dfrac{1}{6P(B)}$

$\rm :\longmapsto\:P(B) = \dfrac{2}{6}$

$\rm :\longmapsto\:P(B) = \dfrac{1}{3}$

Hence, Option (c) is correct.

Additional Information :-

$\boxed{ \rm{ P(A\cup B) = P(A) + P(B) - P(A\cap B)}}$

$\boxed{ \rm{ P(A'\cap B') = 1 - P(A\cup B)}}$

$\boxed{ \rm{ P(A'\cup B') = 1 - P(A\cap B)}}$

$\boxed{ \rm{ P(A\cap B') = P(A) - P(A\cap B)}}$

$\boxed{ \rm{ P(A'\cap B) = P(B) - P(A\cap B)}}$

Answer 2

Answer:

HENCE YOUR ANSWER = 1/3

Step-by-step explanation:

given that P(A) = 1/4

P(A/B) = P(A ^ B) / P(B),,,,,,,,,(1)

P(B/A) = P(B ^ A) / P(A)

here

P(A/B) = 1/2 P(A) = 1/4

P(B/A) = 2/3

HENCE WE CAN WRITE

2/3 = P(B^A) / 1/4

P(B^A) = 2/3*1/4

= 2/12 = 1/6

WE KNOW THAT

P(A^B) = P(B^A)

While substituting in equation 1

1/2 = 1/6 / P(B)

so P(B) = 1/6 / 1/2

P(B) = 2/6 = 1/3