So this is the problem:
An investor is monitoring stocks from Company A and Company B, which
each either increase or decrease each day. On a given day, suppose
that there is a probability of 0.38 that both stocks will increase in
price, and a probability of 0.11 that both stocks will decrease in
price. Also, there is a probability of 0.16 that the stock from
Company A will decrease while the stock from Company B will increase.
What is the probability that the stock from Company A will increase
while the stock from Company B will decrease? What is the probability
that at least one company will have an increase in the stock price?
Things I've written down
If the probability for the price of both company's stock to go up is 0.38 then the probably for this to not happen, will be 0.62 & if this does not happen then would that mean at least one will decrease?
Same thing for the probability for both to decrease since it's .11 then the probability for this not to happen, or in other words for at least one to increase will be .89?
I know the respective answers should be .35 & .89, with .89 being the same as the second thing I wrote down but this seems rather semantic to me.
I can also get the first answer by adding .38+.11+.16 = .65 then 1-.65 = .35 but I can't work out in my head why that would work.
Some help please?
Best Answer
The sum of the probabilities for all possible cases need to add to one.
Since the stocks must go up or down (not stay the same) there are four possible outcomes for two stocks: {$A\uparrow B \uparrow, A\uparrow B \downarrow, A\downarrow B \uparrow, A\downarrow B \downarrow $}.
If you're given three (disjoint) probabilities of the four, subtract from one to get the probability of the fourth: $1 - 0.38 - 0.16 -0.11 = 0.35.$
Your last statement is the correct interpretation because the cases don't overlap at all.