Every trading day

Every trading day, the price of CF Corp stock either goes up by $1 or goes down by $1 with equal likelihood. At the end of 5 trading days, what is the probability that the price of CF Corp stock is up by exactly $3 from its initial price?

(A) 1/16
(B) 1/8
(C) 5/32
(D) 9/32
(E) 3/8

What is the easiest way to solve this problem?

OA C

Hi lheiannie07,
Let's take a look at your question.

The price of CF Corp stock either goes up by $1 or goes down by $1 with equal likelihood, it means that,
$$p\left(Up\right)=\frac{1}{2}$$
$$p\left(Down\right)=\frac{1}{2}$$

We need to find the probability that the price of CF Corp stock is up by exactly $3 from its initial price at the end of 5 trading days.
The price can be $3 at the end of 5 trading days only if during the 4 trading days the price goes up and during on day the price goes down by $1.
For 4 days up the price will be $4 and for 1 day down the price will be $4 - $1 = $3
It means we need to find the probability P(UUUUD) where U represents up price and D represents down price.
Since we have only two outcomes "Up" and "Down", we can solve this using binomial theorem.
$$=\left(nCk\right)p^kq^{n-k}$$
$$=\left(5C4\right)p^4q^{5-4}$$
$$=5\left(\frac{1}{2}\right)^{^4}\left(\frac{1}{2}\right)$$
$$=5\left(\frac{1}{16}\right)\left(\frac{1}{2}\right)=\frac{5}{32}$$

The other ways to find the probability is to find the number of ways , we get 4 ups and 1 down during 5 days. It can be like
DUUUU or
UDUUU or
UUDUU or
UUUDU or
UUUUD or
So there can be 5 ways of getting 4 up days and 1 down day.
We will just multiply this 5 after finding the probability.
$$\Pr\left(Getting\ 4\ Up\ days\ and\ 1\ down\ day\right)=5\times\Pr\left(Getting\ 4\ Ups\right)\times\Pr\left(Getting\ 1\ down\right)$$
$$\Pr\left(Getting\ 4\ Up\ days\ and\ 1\ down\ day\right)=5\times\left(\frac{1}{2}\right)^{^4}\times\left(\frac{1}{2}\right)$$
$$\Pr\left(Getting\ 4\ Up\ days\ and\ 1\ down\ day\right)=\frac{5}{32}$$

Therefore, option C is correct.

Hope it helps.
I am available if you'd like any follow up.