Question

    Two events  and  associated with a sample space of an experiment are such that        ,  and  . Determine whether   and  are independent.

(b)      Two events  and  are such that, they are independent and  and

             and  . Find

i)           the value of

ii)           and  , where  are the complements of events  and  respectively.

Answer 1

(a) Given $E_1$ and $E_2$ are the events that are associated with a sample space of experiment.

$P(E_1) = 0.27\\ P(E_2) = 0.40\\ P(E_1\cup E_2)=0.58$

Condition: For two events $E_1$ and $E_2$ are independent iff

$P(E_1\cap E_2)=P(E_1)\cdot P(E_2)$ , otherwise they are not independent

So, by addition law of probability

$P(E_1\cap E_2)=P(E_1)+P(E_2)-P(E_1\cup E_2)\\$

$=0.27+0.40-0.58\\=0.09$

and $P(E_1)\cdot P(E_2)=0.27\times 0.40=0.108$

So, we can observe that $P(E_1\cap E_2)\neq P(E_1)\cdot P(E_2)$

Hence, we can say that $E_1$ and $E_2$ are not independent events.

(b) Given , two events A and B , such that they are independent

$P(A)=x\\P(B)=x+0.2\\P(A\cap B)= 0.15$

If A and B are independent events, then

$P(E_1\cap E_2)= P(E_1)\cdot P(E_2)\\0.15=x(x+0.2)\\20x^2+4x-3=0$

Upon solving the quadratic equation,

we get $\boxed{x=\dfrac{3}{10}=0.3}$ and x = -0.5 ($x \neq -0.5$ must have a positive value)

option (i) and (ii is not clearly shown in the problem. )