LECTURE 8: Probability (Chapter 13)

A. Overview

B. Basic Definitions

1. The CHANCE of a particular event is the percentage of time that event is expected to occur if the same random process is repeated over and over under the same circumstances. For example, the chance that "a fair die comes up showing 1" is 1/6.

   Note this definition implies the basic process if replicable; if the process cannot be repeated, using the word "chance" is not correct.

2. Chances can take values from 0% to 100%.

3. The chance of a particular event (say rolling a "1" using a fair die) is equal to 100% minus the chance of the event not happening (i.e. 100% minus not rolling a 1).

4. Calculating Basic Probabilities:

          the number of outcomes of interest
          ----------------------------------
          total possible number of outcomes  
   

5. SAMPLING WITH REPLACEMENT -- a situation where the total possible number of outcomes remains the same with every random draw.

6. SAMPLING WITHOUT REPLACEMENT -- a situation where the total possible number of outcomes can change with every random draw.

C. Independence and Conditional Probabilities

1. Independence and Sampling with Replacement (13.4)

   Independence can be thought of as when 2 events occur and one has no effect on the other occurring and vice versa. In other words, (from page 230) if A happens, it does not change the chance of B happening.

   A box with 5 tickets in it labeled 1,2,3,4,5

   (a) if I pull out a ticket, what is the chance that it is a 4? (answer: 1/5)
   (b) if I put the ticket back in and pull another out, what is the chance that it is a 3? (answer: 1/5)

   This is independence. Contrast it with this situation:

2. Conditional Probabilities and Sampling without Replacement (13.2)

   A box with 5 tickets in it labeled 1,2,3,4,5

   (a) if I pull out a ticket, what is the chance that it is a 4? (answer: 1/5)
   (b) if I DO NOT put the ticket back in and then pull another out, what is the chance that it is a 3? (answer: 1/4 note that the chance of has changed) selecting a 3 is not independent. It is dependent where selecting a 4 first had some effect on selecting a 3 afterwards.

   You might think of this situation as selecting a ticket and then not replacing it PUTS A CONDITION on the next ticket selected.

D. Multiplying Probabilities (13.3)

This is a situation where you are trying to figure out the chances of two things happening. It works for both situations of independence and of conditional probabilties.

The chance that two things will happen equals the chance that the first will happen multiplied by the chance that the second will happen given that the first has happened.

In the situation where outcomes are independent, you multiply unconditional probabilities. If the outcomes are dependent (i.e. situations like sampling without replacement) then multiply conditional probabilities.

E. Collins Case (13.5)

This is an example of incorrectly multiplying probabilities.

Several problems here: (a) question the validity of the probabilities (e.g. how do they know that 1/4 of all men have mustaches)? (b) multiplication without consideration of independence can yield extreme results (e.g. are mustaches and beards independent?) (c) All of these calculations assume that you could repeat this situation again and again (e.g. like pulling tickets from a box) Freedman points out that this was something like a unique event.

Bottom line: be careful when multiplying probabilities.

F. Homework

• Exercise Set A: 5 (p. 226)
• Exercise Set C: 7 (p. 230)
• Exercise Set D: 3, 7 (p. 232)
• Review Exercises: 8, 10 (p. 235)