Chapter 4 Introduction to Probability

-in the SR (short-run) events may be viewed as unpredictable, but in the LR (long-run) patterns tend to emerge and we can view a predictable pattern. This is what probability attempts to capture.

A. Probability in General

1. Probability – a numerical measure of the likelihood that an event will occur. It is obtained by measuring a very large number of outcomes.
Notation: P ( event ) =

Properties:
(a) Takes on a value from 0 to 1( 0 means it cannot occur and 1 (or 100% ) means it will occur absolutely and is therefore not by chance.
(b) As we go from 0 to 1 the likelihood of an even occurring is increasing
(c) If something has a probability of 0.5 it is just as likely to occur as to not occur.

2. Experiment (pertaining to probability) – process that generates well-defined probabilities. On any one repetition of an experiment any outcome that is in the sample space may occur.

3. Sample Space – S- set of all possible experimental outcomes. One sample point is one experimental outcome. Generally denoted in some sort of bracket ( {} or [ ].

Ex: roll of a die ( experiment
Sample space ( S = { 1, 2, 3, 4, 5, and 6 } or all possible outcomes that may occur
Sample point ( on individual outcome…say 1
Probability ( since all are equally likely P (event) = 1/6 or for an individual outcome we may write P (rolling a 1) = 1/6

4. Random – when an individual outcomes or outcomes are uncertain and there is ‘regular’ or measurable distribution of outcomes in a large number of repetitions.

5. Multiple Step Experiments – when there is more than one step in the experiment. Before we considered a single step probability, now we look at what happens when there are multiple outcomes we are concerned with.

Ex: flipping two coins or two die instead of one.

a. Counting Rule for Multiple Steps – If an experiment can be described as a sequence of k steps with n possible outcomes with n1 possible outcomes on the first step, n2 possible outcomes on the second step and so one, then the total number of experimental outcomes is given by the expression:

Total outcomes = n1* n2*…….* nk
Example1: Consider flipping two coins. Then we identify the following items; k = 2 for two steps (i.e. two coins) n1 , n2 = 2 for the two outcomes for each (heads and tails)

Total outcomes = 2 * 2 = 4

Example 2: Consider the flip of a coin and the roll of a die. Then we identify the following items; k = 2 for two steps (i.e. one coin and one die roll) n1 = 2 for the two outcomes (heads and tails) n2 = 6 for the six outcomes [ 1, 2, 3, 4, 5, and 6 ]
Total outcomes = 2 * 6 = 12

Note: we still have not assigned probabilities to the individual outcomes that may occur from performing the next experiment. We are simply identifying the total number of outcomes that may occur given certain conditions.

b. Combinations – used when we want to count the number of experimental outcomes from an experiment that involves selecting n objects from a total population of N objects.

[pic]( [pic]) = N! / n! (N-n)!
Where N! = (N)(N-1)(N-2)….(3)(2)(1) n! = (n)(n-1)(n-2)….(3)(2)(1)
0!=1
n < N

example: Consider we check the quality of meat at 3 supermarkets out of a possible 10. What are all the possible combinations we could pick Then N = 10 and n = 3.
[pic] [pic] ( [pic]) = 10! / 3! (10-3)! = 10! / 3!7! (
10*9*8*7*6*5*4*3*2*1 / [3*2*1]*[7*6*5*4*3*2*1] = 10*9*8 / 6 = 120

c. Permutations – This is very much like combinations but now order matters. So even if the outcomes have the same objects, if the order is different then we consider it a different outcome (i.e. if we picked Paul, Sharon, and James in that order and on another turn in the same experiment picked Sharon, Paul, and James…since the order is different even though the outcome has the same results it is considered different from the first). Since this is the case we would expect to have more outcomes and if we look