Introduction to Probability Theory

Computer Lab 2: Finite Sample Spaces and Random Variables

One of the conclusions drawn from Lab #1 was that each outcome in the sample space of an experiment could be assigned a number between 0 and 1, which describes how likely this particular outcome is to occur. This number is the probability of each outcome. In this lab, we further explore this concept on the examples of the rolling of two fair dice and flipping 5 biased coins.

1. Experiment #1

Consider the experiment of rolling two fair dice. We denote each outcome by a pair of numbers corresponding to the numbers turned up by each die. For instance (1,4) means that the first die turned up 1 and the second 4.
• What is the sample space S₁ of this experiment?

  
  

  
  

  
  

  
  

• If the dice are balanced, what is the probability of any particular outcome in the sample space?

  
  

  
  

• Use Calc ->Random Data -> Integer... to to simulate 100 experiments, placing your results in C1 and C2.
• To encode the outcomes, use the Calc ->Calculator... and the expression 10*C1 + C2, storing the results in C3.
• Use Stat->Tables -> Tally... with both counts and percents to estimate the probability of each outcome. Then answer the following questions.
  • Are 100 experiments enough to obtain good estimates of each probability? Why or why not?

    
    

    
    

    
    

    
    

  • How many experiments do you need to get reasonable estimates? How do you know?

    
    

    
    

    
    

    
    

2. Consider the experiment of rolling two fair dice and let X be the random variable that is the sum of the two numbers.
   • Use the computer to simulate 100 roles of the dice in C4 and C5 and the calculator to place the sum in C6.
   • Use the tally command to estimate the probability of each outcome. Plot the results in a graph.

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

   • Can you calculate the probability of each outcome and explain the results of the computer experiments?

     
     

     
     

     
     

     
     

     
     

   • Simulate 10000 rollings of two dice and tally the number of occurrences of each outcome. Then calculate the sum of the numbers obtained on each die and graphs the results. Percentages describing the likelihood of each outcome can then be used to estimate and plot the relevant probabilities.

     
     

3. Consider a binomial random variable with n = 5 and p = 0.75.
   • Simulating Bernoulli trials
     • Use the Calc ->Random Data -> Bernoulli... to to simulate 5 columns of 100 trials, placing your results in C1-C5.
     • Use the Stat -> Tables -> Tally... to check that the percentage of 1's obtained is reasonable.

     
     

   • Simulating a binomial random variable
     • Use the calculator to place the sum C1+C2+C3+C4+C5 in C6. This simulates 100 observations of a binomial random variable with n = 5 and p = 0.75.
     • Use Stat -> Tables -> Tally... to estimate the probability of each event of the form X = x, x= 0, 1,2, 3, 4, 5.
     • Use Graph->Histogram... to produce a plot of the counts for each possible value of X. Such a plot can be used to give you an approximation of the probability function of X.

     
     

   • The distribution of X
     The computer can calculate probabilities Pr (X = x), where X is a binomial random variable. We will use this feature obtain exact values for the probability function 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.

     
     

     
     

     
     

     
     

     
     

     
     

     
     

4. 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?