Question

Select Independent or Not independent for each description of events.

Answer 1

If A and B are independent, then P(A and B) = P(A) * P(B).

Then the probability that A occurs given that B occurs, for example, reduces to the probability that A occurs by itself:

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

So all you need to do is check that P(A|B) = P(A). If so, then A and B are independent. This is true only for the second case.

Answer 2

Answer:

When two events X and Y are independent,

Then, P(X ∩ Y) = P(X) × P(Y)

We know that,

$P(\frac{X}{Y})=\frac{P(X\cap Y)}{P(Y)}\text{ and }P(\frac{Y}{X})=\frac{P(X\cap Y)}{P(X)}$

Thus, by substituting the value of P(X ∩ Y),

$P(\frac{X}{Y})=P(X)\text{ and }P(\frac{Y}{X})=P(Y)$

Which is the required condition for independent events.

In first option,

P(A)=0.3, P(B)=0.5, P(A|B)=0.15

$\implies P(\frac{A}{B})\neq P(A)$

⇒ A and B are not independent events

In second option,

P(A)=0.2, P(B)=0.4, P(A|B)=0.2

$P(\frac{A}{B})=P(A)$

⇒ A and B are independent events.

In third option,

P(A)=0.6, P(B)=0.2, P(A|B)=0.3

$\implies P(\frac{A}{B})\neq P(A)$

⇒ A and B are not independent events