Probability conditional or normal?

Question

I am struggling with this problem.

My work:

I did a) part and I think it should be $$\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)\}$$

for part b) i am not sure will it be $$=1-0.4-0.5=0.1$$

Confused with , and | And I am stumped in part c and onwards!

Answer 1

From the definition of conditional probability, we have

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

Now, here, we have

$$P(I=0|G=0) = \frac{P(I=0,G=0)}{P(G=0)}$$

$$\implies P(I=0,G=0) = 0.4*0.5 = 0.2$$

For the fourth part, you will have to do as above, since you know all the conditional probabilities, and the fact that $P(G=0)=P(G=1)=0.5$

For the 5th part, use $P(I=2 \cap G=1)$ and $P(I=2)$ (just sum over all possible values of G) and it would come from the conditional probability definition

For the last part, you have to establish that

$$P(G=1,I=2) = P(G=1)\cdot P(I=2)$$