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?


  1. What is the probability that you lose all \(20\) games?


  1. What is the probability that you win all \(20\) games?