    ### MIDDLE GROUND - Basic Probability and Counting Problems

Basic Probability
Vocabulary
Facts
Number of Ways to Make An Ordered List Or A Group
The average is the sum of the products of the event and the probability of the event.

Basic Probability

Vocabulary
experiment -- an action that is performed
event -- a result of an experiment or action, an outcome
outcome -- a result of an experiment or action, an event
sample space -- the set of all possible outcomes or events
probability of an event -- P(event), or p(event)
ex. p(A), the probability of event A
ex. p(x=3), the probability the variable is 3 or
the probability of the number 3 occuring,
or the probability of obtaining a 3.
probability of an event, P(event) = f/n,
 (frequency of the event) P(event) = (number of events in the sample space)

Facts
The lowest possible probability an outcome might have is 0. If P(event)=0, the event can not happen.

The highest possible probability an outcome might have is 1. If P(event)=1, the even does happen.

Probabilities range between 0 and 1, inclusive.
0 < P(event) < 1.

The sum of all the probabilities for an experiment is 1.
ex. Experiment: flip a fair coin.
ex. Experiment: pick a day of the week.
p(Sunday) + p(Monday) + ... + p(Saturday) = 7/7 = 1
ex. Experiment: pick a day of the week
p(January) = 0

The Expected Value is the Mean.
One way of describing the results of an experiment is to state the value you expect to get. In any experiment the mean, the arithmetic average, is the expected value.

Should you be surprised if when you complete the experiment you don't get 2 heads? No. The expected value is kind of an average. Notice that 10 times out of the 16 possible events in the sample space the number of heads is not 2. The mean, called the expected value, is however, 2. Notice that, in the experiment given above, the mean is 2. The mode and the median are also 2. Though the definition of expected value is the mean, each of the everyday averages -- mean, mode, and median -- in this case is 2.

Number of Ways to Create An Ordered List or a Group

These are counting problems. They are used here as an intro to important ideas in probability. They should be considered before reading the page Binomial Distribution.

1. A club has 4 members and wishes to elect a president, vice president, secretary, and treasurer. In how many ways can this be done?
There are 4 ways to select a president, but then only 3 ways to choose the vp, but then only 2 ways to choose the secretary, and then only 1 way to choose the treasurer,

so, there are 4(3)(2)(1) or 24 ways to do this.

Using members A, B, C, and D, here is a tree diagram and the sample space with 24 ways. 2. A club has n members and wishes each member to hold an elected office, beginning with the president, vice president, secretary, and treasurer. In how many ways can this be done?
The answer is n factorial ways. There are n ways to fill the first office, n-1 ways to fill the 2nd office, n-2 ways to fill the 3rd office, and so on. There are n! or n(n-1)(n-2)(n-3) ... (3)(2)(1) ways.

3. A club has 4 members and wishes to elect a president and then a vice president. In how many ways can this be done?
There are (4)(3) or 12 ways to do this. Here order counts. A list is required. For a source set of n elements, if order is important and x elements are chosen for the ordered list, a permutation of n things taken x at a time is required, Pn,x. There are n! / (n -x)! ways to do this.
 n! Pn,x = (n-x)!
 n! (number of ordered lists given n things) Pn,x = = (n-x)! (shorten the list so only x things are used)

In this problem, an ordered list of 4 elements, take 2 elements is required, P4,2. There are 4! / (4-2)! = 24/2 = 12 ways to do this.
 4! 4(3)(2)(1) P4,2 = = = 12 (4-2)! (2)(1)

4. A club has 4 members and wishes to form a committee of 2 member. In how many ways can this be done?
This time order does not count, so, AB is really BA and should not be counted as different committee.
This time a combination is used. A combination is a permutation divided to remove duplicates. n! (number of ordered lists given n things) Cn,x = = (n-x)! x! (shorten to only x things)(remove duplicates)

 4! (4)(3)(2)(1) C4,2 = = = 6 (4-2)! 2! (2)(1)(2)(1)
5. A club has n members and wishes to form a committee of r member. In how many ways can this be done?
Order does not count. Use a combination.

 n! Cn,x = (n-x)! x!

The average is the sum of the products of the event and the probability of the event.

In the example below, the task is to compute the average of test grades. These steps were chosen to complete the task, but, also to illustrate where the definition of the average as the sum of the products of the event and its probability is related to work already familiar to the reader. The most important work focuses on the fact that rather than add individual terms, the terms may be grouped by frequency. When the frequency is divided by the number of things to be averaged, a probability is obtained. The probability of an event is the frequency divided by the number of events in the sample space, P(event) = f/n,

 (frequency of the event) P(event) = (number of events in the sample space)

This definition of average, or expected value, may seem of little value here, but, the number in the sample space, the number of tests to be averaged, is only 5. Sometimes the size of the sample space is very large. Sometimes the size of the sample space is not given. Yet, the average or expected value of a probability distribution may be computed using the formula, sum of the products of the event and the probability of the event. Why the mean, expected value, is (p)(n)         What follows is the formal justification of the formula for the mean given above -- the mean, expected value, is the sum of the products of the event and the probability of the event.        © 2010, Agnes Azzolino www.mathnstuff.com/math/spoken/here/2class/90/basic.htm