Question

If sample space S={e1,e2,e3,e4}, find P(e1 ) and P( e2) if P(e3)=1/3 and P(e2)=3P(e1)

Answer 1

$S=\{ e_1,e_2,e_3,e_4\}$

Given,

$P(e_3)=\dfrac{1}{3}\\P(e_2)=3P(e_1)$

Now,

$P(e_1)+P(e_2)+P(e_3)+P(e_4)=1\\\implies P(e_1)+P(e_2)+P(e_4)=1-\dfrac{1}{3}=\dfrac{2}{3}\\\ \\\implies P(e_1)+3P(e_1)+P(e_4)=\dfrac{2}{3}\ [\because P(e_2)=3P(e_1)]\\\ \\\implies 4P(e_1)+P(e_4)=\dfrac{2}{3}$

Since, the value of $P(e_4)$ is not given , so we assume the value of $P(e_4)$

Let $P(e_4)=\dfrac{1}{6}$

So,

$\implies 4P(e_1)=\dfrac{2}{3}-\dfrac{1}{6}=\dfrac{3}{6}=\dfrac{1}{2}\\\ \\\implies P(e_1)=\dfrac{1}{8}$

and

$P(e_2)=3P(e_1)\\\implies P(e_2)=\dfrac{3}{8}$