Question

if a and b are events such that p(AuB)=5/6,p(A^)=(1/4),p(B)=1/3 then A and B are
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Answer 1

Given,

$\longrightarrow\sf{P(A')=\dfrac{1}{4}\implies P(A)=1-\dfrac{1}{4}=\bold{\dfrac{3}{4}}}$

$\longrightarrow\sf{P(B)=\dfrac{1}{3}}$

$\longrightarrow\sf{P(A\cup B)=\bold{\dfrac{5}{6}}}$

We know ,

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

$\implies\sf{P(A \cap B)=\dfrac{3}{4}+\dfrac{1}{3}-\dfrac{5}{6}}$

$\implies\sf{P(A \cap B)=\dfrac{9}{12}+\dfrac{4}{12}-\dfrac{10}{12}}$

$\implies\sf{P(A \cap B)=\dfrac{9+4-10}{12}}$

$\implies\sf{P(A \cap B)=\dfrac{3}{12}}$

$\implies\sf{P(A \cap B)=\dfrac{1}{4}}$

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$\sf{P(A)\times P(B)=\dfrac{3}{4}\times \dfrac{1}{3}}$

$\implies\sf{P(A)\times P(B)=\dfrac{3}{12}}$

$\implies\sf{P(A)\times P(B)=\dfrac{1}{4}}$

$\implies\sf{P(A)\times P(B)=P(A\cap B)}$

∴ A and B are independent but not equally likely.

Answer 2

Answer:

They are independent

P(A) × P(B)= P(A intersection B)