Probability Models

We have already learned how to used models for linear relationships and for some normal distributions in the normal density curve. Now we create a mathematical description or model for randomness. First think about a very simple random phenomenon, tossing a coin once. We describe the coin toss in two parts: a list of possible outcomes and a probability for each outcome. The list of all the possible outcomes is called the SAMPLE SPACE (S). An event is any outcome or set of outcomes of a random phenomenon. An event must be present in the sample space. A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space (S) and a way of assigning probabilities to events.

To specify S we must state what constitutes an individual outcome and then state which outcomes can occur. We have some freedom in defining the sample space and make choices through convenience. Being able to properly enumerate the outcomes in a sample space will be CRITICAL to determining probabilities. Creating a "tree diagram", as used in elementary school, can be helpful with all of its branches labeled. Learning the MULTIPLICATION PRINCIPLE for events occurring is also helpful. The rule states:

If you can do one task in "x" number of ways and a second task in "y" number of ways, then BOTH tasks can be done in x times y number of ways. Try this yourself before seeing the explanation in the book.

An experiment consists of flipping four coins. You can think of either tossing four coins on the table at at once OR flipping a coin four times in succession and recording the four outcomes. One possible outcome is HHTH. Because there are two ways each coin can come up, the multiplication principle says that the total number of outcomes is
2 X 2 X 2 X 2 =16. See if you can list all the possibilities...try to describe your method.

It is not a huge step from tossing coins to polling four people at random regarding whether they favor reducing federal spending on low-interest student loans. Each has only two possible outcomes. Even when the numbers are increased to 1500 or some such for the poll, it is analogous to 1500 coin tosses. ONE OF THE GREAT ADVANTAGES OF MATHEMATICS IS THAT THE ESSENTIAL FEATURES OF QUITE DIFFERENT PHENOMENA CAN BE DESCRIBED BY THE SAME MATHEMATICAL MODEL. Read the last sentence again and again until you grasp its importance. (Remember all the different types of word problems that resulted in linear models, or quadratic models, or exponential modes from Algebra 2???)
In the case of coin tossing or dice rolling, the sample space has a finite number of outcomes possible. Sometimes the sample space is infinite as when a computing system has a function that generates a random number between 0 and 1. S is a mathematical idealization and in this case would best be described using an interval.

IF you select random digits by drawing numbered slips of paper from a hat, and you want all ten digits to be equally likely to be selected each draw, then after you draw a digit and record it, you MUST put it back into the hat...then the second draw will be exactly like the first. This is called sampling WITH REPLACEMENT. IF you do NOT replace the slips you draw, there are only nine choices for the second slip picked, and eight for the third and so on. This is called sampling WITHOUT REPLACEMENT. So...if the question is "How many
3-digit numbers can you make?" the answer is 10 X 10 X 10 = 1,000 ways by the multiplication principle PROVIDING all ten numbers are eligible for each of the three positions in the number. On the other hand, there are 10 X 9 X 8 = 720 different ways to construct a three-digit without replacement. You should be able to determine from the context of the problem whether the selection is with or without replacement and this will help you identify the sample space.

Probability has specific rules and notation that will guide us with more complex situations:
1. any probability is a number between 0 and 1. An event with probability = 0 NEVER occurs, and an event with probability 1 occurs on every trial. Symbolically, 0 < P(A) < 1 where A is an event and P(A) is the probability of that event occurring.
2. All possible outcomes together must have probability 1...the sum of the probabilities for all possible outcomes must be exactly 1. All possible outcomes is another way to say sample space, so symbolically P(S) = 1.
3. The probability that an event does NOT occur is 1 minus the probability that the event DOES occur. The probability that an event OCCURS and the probability that is does NOT occur always add to 100%, or 1.
Symbolically, P(A^c) = 1 - P(A) where A^c refers to the complement of A which is A does not occur.
4. If two events have no outcomes in common, the probability that one OR the other occurs is the SUM of their individual probabilities. Symbolically, P(A or B) = P(A) + P(B) iff the events are disjoint or mutually exclusive. Disjoint means have nothing in common.

The use of Venn diagrams and the use of set notation is very helpful for picturing and talking about events occurring. The event {A U B} read "A union B", is the set of all outcomes that are EITHER IN A OR IN B. This is another way to say "OR" like rule #4 above. The symbol 0 is used for empty event. If two events A and B are disjoint, we write "A intersect B" is empty. Sometimes we emphasize that we are describing a compound event by enclosing it within braces. In Venn diagrams there will be NO overlap of the A and B event areas. Some other logical conclusions are "A union A^c" = S and "A intersect A^c" = empty set.

Sec. 6.2(b)
Index