2.5 Applications of Binomial distribution

• You can model many complex business problems by using probability distributions. These distributions provide answers to questions such as: “What is the likelihood that oil prices will rise during the coming year?”, “What is the probability of a stock market crash next month?”, or “How likely is it that a corporation’s earnings will fall below expectations this year?”

• To answer aforementioned questions an appropriate distribution for a given problem should be considered

• Some of the more widely used probability distributions in business are the Binomial and Poisson distribution of discrete random variable (only a countable number of values are possible)

• The Binomial distribution computes the probabilities of events where only two possible outcomes can occur (success or failure), e.g. when you look at the closing price of a stock each day for one year, the outcome of interest is whether the stock price increased or not.

• Probability that a specified number of successes will occur during a fixed number of trials is calculated by Binomial formula:

$$\begin{align} P(X=x_i)&=\frac{n!}{x_i!(n-x_i)!}p^{x_i}(1-p)^{n-x_i}\\ \\ x_i&=\text{the number of successes } (x_i:~0,~1,~2,...,n) \\ \\ n&=\text{the number of trials} \\ \\ p&=\text{the probability of success on a single trial} \\ \\ (1-p)&=\text{the probability of failure on a single trail} \\ \\ \frac{n!}{x_i!(n-x_i)!}&=\text{the number of combinations without replacement}\\ \tag{2.10} \end{align}$$

• If a random variable $X$ follows a Binomial distribution we use notation

$$\begin{equation} X \sim B(n,~p) \end{equation}$$

• The expected value of the Binomial distribution is

$$\begin{equation} E(X)=np \end{equation}$$

• The variance of the Binomial distribution is

$$\begin{equation} Var(X)=np(1-p) \end{equation}$$

• To compute Binomial probabilities in Excel you can use function =BINOM.DIST(x;n;p;FALSE) with setting the cumulative distribution function to FALSE (last argument of the function)

Example 2.6 Suppose you play a game that you can only either win or lose. The probability that you win any game is $55\%$, and the probability that you lose is $45\%$

1. What is the probability that you win $15$ times if you play the game $20$ times?

$P(X=15)=$_______

2. What is the probability that you lose all $20$ games?

$P(X=0)=$_______

3. What is the probability that you win all $20$ games?

$P(X=20)=$_______