Question

Suppose that the increase or decrease in the price of a stock between the beginning and the end of a trading day is considered an equally likely event. What is the probability that a stock will:
a) increase on one out of four trading days?
b) decrease on two out of four trading days?
c) increase on not more than two of four trading days?
d) decrease on at least three of the four trading days?

Answer 1

The probability of  increase or decrease is $p=0.5$ .

The the number of increases X out of 4 (in 4 days) follows Binomial distribution with $n=4,p=0.5$
The PMF of X is $P\left ( X=x \right )=\binom{n}{x}p^x\left ( 1-p \right )^{n-x},x=0,1,2,...,n$ .

a) The probability that a stock will increase on one out of four trading days is

$P\left ( X=1 \right )=\binom{4}{1}0.5^1\left ( 1-0.5 \right )^{4-1}\\ {\color{Blue} P\left ( X=1 \right )=0.25}$

b)The probability that a stock will decrease on two out of four trading days is

$P\left ( X=2 \right )=\binom{4}{2}0.5^2\left ( 1-0.5 \right )^{4-2}\\ {\color{Blue} P\left ( X=2 \right )=0.375}$

c)The probability that a stock will increase on not more than two of four trading days is

$P\left ( X\leqslant 2 \right )=\sum_{x=0}^{2}\binom{4}{x}0.5^x\left ( 1-0.5 \right )^{4-x}\\ {\color{Blue} P\left ( X\leqslant 2 \right )=0.6875}$

d)The probability that a stock will decrease on at least 3 of four trading days is

$P\left ( X\geqslant 3 \right )=1-P(X\leqslant 2)\\ P\left ( X\geqslant 3 \right )=1-0.6875\\ {\color{Blue} P\left ( X\geqslant 3 \right )=0.3125}$