# Probability of one stock price rising, given probabilities of several prices rising/falling

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.

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.

The 4 Events

• $E_1$ = Both decrease
• $E_2$ = Both increase
• $E_3$ = A increases, B decreases
• $E_4$ = A Decreases, B increases

compliment each other in the sense that exactly one of these events will happen. Therefore we have

$$P(E_i\cap E_j)=0, i\neq j$$

and

$$P(E_1)+P(E_2)+P(E_3)+P(E_4)=1$$

What is the probability that the stock from Company A will increase while the stock from Company B will decrease?

Since you have the provavilities of $E_1,E_2$ and $E_4$ given, you can calculate

$$P(E_3)=1-P(E_1)-P(E_2)-P(E_4)=0{,}35$$

What is the probability that at least one company will have an increase in the stock price?

Unless $E_1$ happens, one of the companies will ahave an increase in the stock prices. The solution for the second question is therefore

$$1-P(E_1)=0{,}89$$

• just posted the exact same answer. Removed it as you were first. Jan 27, 2015 at 23:53

To summarize:

$P($Both Stocks Go Up$) = 38\%$
$P($Both Stocks Go Down$) = 11\%$
$P($A Down, B Up$) = 16\%$
$P($A Up, B Down$) = ?$

Notice that these 4 events make up the entire set of possible combinations of the movement of stocks $A,B$. Because of this, their probabilities add to $1$, and we can conclude
$P($A Up, B Down$) = 1 - (.38 + .11 + .16) = 1 - .65 = .35$

The likelihood that at least one goes up is $1 - P($Both Go Down$) = 1 - .11 = .89$