5.3 Binomial distribution - university of calgary - base content  (Page 2/30)

Notation for the binomial: B = binomial probability distribution function

X ~ B ( n , p )

Read this as " X is a random variable with a binomial distribution." The parameters are n and p ; n = number of trials, p = probability of a success on each trial.

Binomial formula

P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$

• $p^x$ is the probability of x successes in x independent and identical trials. For example, if the probability of success = 0.4, the probability of five successes in five independent and identical trials is
  $0.4\times 0.4\times 0.4\times 0.4\times 0.4=\mathrm{0.4}^5=p^x$
• $q^{\mathrm{n-x}}$ is the probability of n - x failures in n - x identical and independent trials. For example, if the probability of success = 0.4, then the probability of failure is 1 - p = 0.6. If there are eight trials ( n = 8) with five successes ( x = 5 ), then there were three failures in the eight trials ( n - x = 8 - 5 = 3). The probability of three failures in three independent and identical trials is
  $0.6\times 0.6\times 0.6=\mathrm{0.6}^3=q^{\mathrm{n-x}}$
• $\left(\begin{array}{c}n\\ x\end{array}\right)$ represents the number of combinations of x successes in n trials. If there are eight trials ( n = 8) and five successes ( x = 5), then there are 56 possible ways to arrange five successes among eight trials.
  $\left(\begin{array}{c}8\\ 5\end{array}\right)=\mathrm{56}$
• The formula P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$ is the probability of x successes in n independent and identical trials. If there are eight independent and identical trials, the probability of five successes where p = 0.4 is
  $P(\mathrm{X=5})=\left(\begin{array}{c}8\\ 5\end{array}\right)\mathrm{0.4}^5\mathrm{0.6}^3=\mathrm{0.1239}$