Introduction to Probability Theory

Computer Lab 3: The Binomial Random Variable

In this lab, we will first use the computer to both estimate and calculate the probability mass function of a binomial random variable. We will then plot its distribution function and solve some word problems.

1. The binomial random variable

Consider a binomial random variable with n = 5 and p = 0.75.
• Simulating Bernoulli trials
  • We will simulate a Bernoulli trial by asking the computer to choose a number at random from the following list: 1, 1, 1, 0. Note that the probability of obtaining a 1, which we will call success, is p = 0.75. To do so, follow the steps below.
    1. Enter the above list in Column C1.
    2. Use Random Data -> Sample From Columns... in the Calc menu. Enter the number of samples you want to create (i.e. the number of Bernoulli trials you want to perform), indicate you are sampling from Column C1 and that you want to store the results in Column C2. Make sure the Sample with replacement box is checked.
  • Use the Stat -> Tables -> Tally... to check that the percentage of 1's obtained is reasonable. Increase the number of trials if necessary.
  • Repeat the experiment with a different list (e.g. with 2 0's and 6 1's) to convince yourself that the results are consistent.

  
  

• Simulating a binomial random variable
  • Create 5 (i.e. the value of the parameter n of the binomial random variable) columns of samples from Column C1.
  • We will consider that each row of 5 numbers represents 5 consecutive Bernoulli trials. Calculate the value of X, the binomial random variable with n = 5 and p = 0.75, for each of these sets of trials.
  • Use Stat -> Tables -> Tally... or Stat -> Basic Statistics -> Store Descriptive Statistics... to estimate the probability of each event of the form X = x_k.
  • Use the Graph menu to produce a plot of these probabilities as functions of each possible value of X. Such a plot gives you an approximation of the probability mass function (p.m.f.) of X.

  
  

• The distribution of X
  The computer can calculate probabilities Pr (X = x_k), where X is a binomial random variable. We will use this feature obtain exact values for the p.m.f. of X with n = 5 and p = 0.75.
  • Enter all possible values that X can take in say Column C8.
  • Use Calc -> Probability Distributions -> Binomial to calculate the exact probabilities.
    1. Choose to calculate a Probability
    2. Enter 5 as the Number of trials and 0.75 as the Probability of success.
    3. Use Column C8 as the Input column and enter the column where you want the probabilities to be written in the Optional strorage field.
    4. Then click OK
  • Plot the exact probability mass function of X and compare the result with the approximate graph you had before.
  • The (cumulative) distribution function (c.d.f.) F of X is defined by
    F(x) = Pr (X ≤ x)
    Use the plot of the probability mass function of X to draw the graph of F below

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    
    

    Check your answer by using the computer to plot a graph of F. You can calculate probabilities of the form Pr (X ≤ x) by using Calc -> Probability Distributions -> Binomial and choosing the Cumulative Probability option.
2. Word Problems

From A First Course in Probability by Sheldon Ross, Prentice Hall, 2002.
1. A communication channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability 0.2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0 and 11111 instead of 1. If the receiver of the message uses “majority” decoding, what is the probability that the message will be wrong when decoded? What independence assumption are you making?

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

2. A student is getting ready to take an important oral examination and is concerned about the possibility of having an “on” day or an “off” day. He figures that if he has an on day, then each of his examiners will pass him independently of each other, with probability 0.8, whereas, if he has an off day, this probability will be reduced to 0.4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student feels that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?