Introduction to Probability Theory

Computer Lab 2: Finite Sample Spaces

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 example of the rolling of two dice.

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 the computer to simulate 100 experiments, and use the results 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. Experiment #2

Consider the experiment of rolling two fair dice and adding up the two numbers.
• What is the sample space S₂ for this experiment? Outcomes in S₂ correspond to events in S₁. Explain.

  
  

  
  

  
  

  
  

• Use the computer 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?

  
  

  
  

  
  

  
  

  
  

3. Reading assignment: Section 1.6