An investment company estimates that the probability that stocks a and b go up is 61 % and 75%,
respectively. Let A be the event that stock a goes up, and let B be the event that stock b goes up.
Assume for simplicity that a stock can only go up or go down (i.e. ignore the case where the stock
maintains exactly the same value).
(a) Assuming that the events A and B are independent, compute the probability that both
stocks go up.
(b) Show that the probability that both stocks go up cannot be smaller than 36%, no matter
whether events A and B are independent or not.
(c) Suppose that it is impossible for both stocks to go down. Compute the probability that
both stocks go up.
So I am dealing with question of probability, I manage to figure the a) and b) sub question by understand the meaning of dependent event and independent event but I am kind of the stuck in the third sub question c)
a) is simply multiply 61% and 75% together
b) is the probability the stock a goes up multiply the probability the stock b goes up given that stock a goes up
How about c)?
I am guessing that I need to use conditional probability with the statement (impossible for both stock to go down) but I have no idea how to translate the statement in to math
Best Answer
For both $(b)$ and $(c)$ parts, use the inclusion-exlcusion principle:$$P(A\text{ and }B)=P(A)+P(B)-P(A\text{ or }B)$$For $(b)$, note that $P(A\text{ or }B)\le1$ so $P(A\text{ and }B)\ge P(A)+P(B)-1=0.36$.
For $(c)$ you are given $P(A'\text{ and }B')=0\implies P(A\text{ or }B)=1-P(A'\text{ and }B')=1$ giving $P(A\text{ and }B)=0.36$.