Introduction to Probability Theory

Lab 1: Experiments, Sample Space, Events

Equipment: a bag containing colored objects of similar shape. DO NOT LOOK INSIDE THE BAG.

1. Experiment:
We call experiment a process whose outcome is not known in advance with certainty. For instance, perform the experiment of drawing an object from the bag and recording its color. Indicate the outcome in the table below and return the object to the bag.

Experiment #	Outcome
1	...

2. Sample space: The possible outcomes of the above experiment are the different colors, Yellow, Blue, Red, Pink, Black, White, Brown, Green, Orange of the cubes. The collection of all of these outcomes is called the sample space of this particular experiment. Draw your own representation of the sample space below.
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

3. Events: An event is a collection of outcomes, or a subset of the sample space. In particular, the whole sample space, as well as the empty set are events. Represent the event corresponding to the outcome of Experiment #1 above in your sketch of the sample space.

   
   
   
   

4. Estimate the contents of the bag
   
   • Perform 20 experiments, each of which consists in drawing an object from the bag, recording its color and replacing the object into the bag. Record the outcomes in the table below:

     
     
     
     

     Experiment #	Outcome	Experiment #	Outcome
     2	...	12	...
     3	...	13	...
     4	...	14	...
     5	...	15	...
     6	...	16	...
     7	...	17	...
     8	...	18	...
     9	...	19	...
     10	...	20	...
     11	...	21	...

   • Use the above information to estimate the contents of the bag. You can try to guess how many objects are in the bag, but do not look yet.
     

     
     

   • Look at the contents of the bag. How good was your estimate?
     

     
     

   • Are you surprised by the contents? Why or why not?
     

     
     

   • Now that you know the contents of the bag, give the number of each color that you expected to draw. How far away from that number would you guess is a typical deviation?
     

     
     

   • What could you have done to improve your estimation of the contents of the bag? Would your strategy depend on the number of objects in the bag? Why or why not?
     

     
     

   • Find the others in the class that had the same contents as you and see if your estimate for the contents of the back improves if you aggregate the outcomes of your experiments.
     

     
     

5. Reading Assignment: Sections 1.1 to 1.4