Theoretical vs. Empirical Probabilities

[Pages:15]MAT 121

Spring 2013

Fisher

In the previous class, we discussed a wide variety of topics regarding probability:

Theoretical vs. Empirical Probabilities o Theoretical Probability: o Empirical Probability:

Disjoint/Mutually Exclusive Events (or) o

Independent Events (and) o Important: If the second event is not independent then we must adjust the probability accordingly (for instance, if there is replacement versus when there is not replacement)

Conditional Probability o

One final IMPORTANT piece of the probability puzzle is the ability to count potential outcomes. Especially when talking about theoretical probabilities, potential outcomes are literally half the equation.

There are multiple counting techniques and which one we should use depends entirely on several factors. Some of those factors include whether repetition is allowed or whether order matters. We will now explore all the counting methods and how to tell when to use them.

The first technique we will discuss is the most basic and fundamental method, it is actually referred to as the Fundamental Counting Principle. The Fundamental Counting Principle, FCP, states that to find all possible outcomes you simply multiply the possible choices for each event. The product of this process represents the total possible outcomes or arrangements.

For example, if a menu gave you a choice of 3 appetizer, 4 entrees, and 5 desserts, then there are:

Sections Covered: 5.5; 6.1-6.3

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Spring 2013

Fisher

The text will refer to this as the Multiplication Rule of Counting, stating that if you have p selections

for the first choice, q selections for the second choice, r selections for the third choice, and so on, then

the task of making these selections can be done in

different ways.

There are many ways you can be asked to utilize this principle, below are a few samples of how the multiplication rule of counting can be implemented.

Example 1: Employees at a local business are issued employee ID's consisting of 2 letters and 3 numbers. The letters may repeat however the numbers may not. How many different employee ID's are possible using this system?

Note: In this example, some repetition was allowed (for the letters) but some choices could not be used more than once (for the numbers). When using the multiplication rule of counting, you must pay attention to whether repetition is allowed since that will impact how many choices you have for subsequent selections.

Example 2: A student body council must select a president, vice-president, and secretary for their group. If there are 50 members of the council, how many different ways could the officers be selected?

Sometimes when using the multiplication rule of counting you will be asked to use all the choices. For instance, if I wanted to arrange 6 textbooks on a shelf, how many ways could I position them?

You would have 6 choices for the first position, 5 choices for the second position, 4 choices for the

third, 3 choices for the fourth, 2 choices for the fifth, and lastly only 1 choice for the last position.

Using the rule, we would multiply

. If we know we need to use all

the choices, we can save time using a special symbol known as the factorial symbol (n!). Using the

factorial symbol (!), we know to multiply the number n by all values in descending order down to 1.

Using the example from above regarding the 6 textbooks, we really were calculating:

Sections Covered: 5.5; 6.1-6.3

MAT 121

Spring 2013

Fisher

An example that utilizes factorial:

A traveling sales representative must travel to 8 different locations to service existing accounts. How many different routes are possible?

As discussed above, sometimes repetition is not allowed. There is a special name for the multiplication

rule for counting when it involves making distinct selections, this name is permutations. The book

defines permutations as "an ordered arrangement in which r objects are chosen from n distinct objects

so that

and repetition is not allowed."

The symbol for permutation is nPr. There are 3 requirements to consider when using the permutation function:

1) Are the objects distinct? 2) Is repetition prohibited? 3) Is order important?

If all 3 answers are yes, then we use the permutation function: nPr =

Before we use it to count possible outcomes, let's practice using the formula first.

Calculate 5P3

5P3 =

Note: We stop expanding at 2!. Why?

Calculate 6P6 6P6 =

Important!

Sections Covered: 5.5; 6.1-6.3

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Spring 2013

Fisher

Now let's use permutations within the context of a counting outcomes problem. Example: Are you feeling lucky??? In horse racing, one can bet on which horses will finish in 1st, 2nd, and 3rd. also known as the trifecta. If there are 10 horses in the race, how many ways can the first 3 positions be filled? Since order matters in a race, there can be no repeats (a horse can't get 1st and 2nd place), and there are 10 different horses racing, this is a perfect time to use a permutation.

There are 10 choices (n) and we want to pick 3 (r), so our permutation will be:

10P3 =

So there are 720 different ways that 10 horses could finish in the first 3 positions. No one said gambling was easy!

Try this! Going one step further with the previous example, a perfecta is a term used in horse racing where a gambler attempts to pick who the top 4 finishers will be. How many ways can a 10 horse race finish for the top 4 horses?

10P4 =

You can start to see why this is so difficult to do, however if you can catch lightning in a bottle you could become quite wealthy!

DISCLAIMER!!! I DO NOT IN ANY WAY ENCOURAGE STUDENTS TO PARTICIPATE IN GAMBLING LEGAL OR OTHERWISE!!!

Let's think back to the study body council from above, instead this time let's choose 5 members to serve on a fundraising committee. Does order matter anymore? The answer is no. It doesn't matter what order I choose the 5 members since there is no inherent difference between committee members in terms of power or responsibilities. Since order no longer matters, we must find another method (permutation cannot be used if order does not matter) for determining the total number of ways a 5 member committee could be chosen. Enter combinations! Combinations are very similar to permutations (there must be n distinct objects and repetition is not allowed) with one significant difference, order no longer matters.

The formula for combinations is nCr =

Note: Look how similar the formulas are!

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Let's just practice calculating a combination. Calculate 6C2 6C2 =

Fisher

Calculate 8C8 8C8 =

Let's put combinations into context.

Example A local charity has raised funds to send some local honor society students to Washington, D.C. for a national honor society convention. There are 20 members in the local chapter and they must select 5 members to send. How many different group arrangements are possible?

20C5 =

One of the skills we worked on earlier this semester was selecting samples from populations. Using combinations, you can determine how many samples of size n can be selected from a population of size N. For instance, how many samples of size n = 3 can be selected from a population of 15 individuals?

15C3 =

Up to this point we have made one common assumption for both permutations and combinations (and the multiplication rule for counting in general), that all choices involved were distinct or all different. What happens when we take this assumption away, what if we do have choices that repeat?

Think about this: How many ways could the letters in the word STATISTICS be rearranged?

Sections Covered: 5.5; 6.1-6.3

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Spring 2013

Fisher

The problem here, unlike in the previous counting methods, is there are letters that are used more than once. The letter S appears 3 times, T 3 times, and I twice. So if I randomly choose a T, which T did I pick? This is what is meant by nondistinct objects.

So if I want to rearrange the letters in the word STATISTICS, we must consider each letters potential sequence within the entire group of choices.

So there are:

10C3 ways of positioning the S's o (Notice we use a combination since we can't tell the difference between S's)

7C3 ways of positioning the T's o (Notice the number of choices, n, is decreasing since we have less positions left)

4C2 ways of positioning the I's 2C1 ways of positioning the A's 1C1 ways of positioning the C's

So we would "simply" multiply: (10C3) (7C3) (4C2) (2C1) (1C1) =

Easy right??? With a few cancellations the product becomes nicer, direct approach, but first let's finish the calculation:

. We will discuss a more

We used combinations above because the choice of the each letter (especially the nondistinct ones) didn't matter since we could not tell the difference between each of the S's, T's, or I's. Overall though, the order of how we arrange the letters DOES matter and therefore this is a permutation. Luckily there is a formula for permutations with nondistinct objects:

The number of permutations of n (nondistinct) objects =

Try This! How many ways can the letters in the word MISSISSIPPI be rearranged? There are 11 total objects (n) with: 1 M (n1), 4 I's (n2), 4 S's (n3), and 2 P's (n4).

Total Arrangements =

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Let's put all this counting techniques to use, let's calculate some probabilities.

Fisher

Winning the New York State lottery, everyone's dream and the easy way to the high life. But what are the chances? You may have been told that you have a better chance of being struck by lightning. According the National Weather Service, the chances of being struck by lightning in any given year are

. These are pretty long odds and therefore highly unlikely. On a side note, there is

approximately a

chance of being struck by lightning in your lifetime and scarier yet odds of

knowing someone who will be struck. That means, statistically speaking, if you have 1,000 Facebook friends that 1 of them will get struck in your lifetime! Yikes!!!

Back to the problem at hand, is winning the NY Lottery less likely than getting struck by lightning during any given year?

First we must determine how many ways are there of having the 6 numbers on your ticket. There are 59 numbers in play during a lottery drawing and since the order of the numbers doesn't matter we will calculate the combination 59C6.

59C6 =

Since you get two plays for $1, the number of ways of achieving success is 2. In other words, you have two chances per ticket.

This is approximately

. In summary, 1 out of 1 million people get struck by

lightning while 1 out of 25 million wins the lottery. Your odds of getting struck by lightning are WAY better than winning the lottery. As a matter of fact, you are almost more likely to get struck by lightning AND know someone else who got struck by lightning than you are to win the lottery. But someone has to win, $1 and a dream! Again, see disclaimer above.

Sections Covered: 5.5; 6.1-6.3

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Spring 2013

Fisher

We now move into chapter 6. We will review some terminology from earlier this semester.

Discrete Continuous Random

Using these ideas, we will now expand on them. A random variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. A random variable can be either discrete or continuous. A discrete random variable is a random variable that has either a finite or countable number of values. A continuous random variable is a random variable that has infinitely many values.

A discrete random variable can be plotted on a number line with spaces between each point whereas a continuous random variable can be plotted on a number line in an uninterrupted way.

If I rolled a standard number cube 5 times and the random variable X represents the number of 3's that resulted from the rolls, then the possible values of X are x = 0, 1, 2, 3, 4, or 5. Therefore X in this case would be a discrete random variable.

If I was measuring the wait time of the next car to pull through the drive-thru window at a fast-food restaurant and the random variable X represents the wait time, then the possible values of X are infinite (x > 0 minutes). Therefore X in this case would be a continuous random variable.

We will spend the majority of our remaining time focusing on the discrete random variable.

A probability distribution is similar to a frequency distribution. A probability distribution of a discrete random variable lists all possible values of the random variable along with their corresponding probabilities. Like frequency distribution, probability distributions can be modeled in the form of a table, graph, or formula.

Sections Covered: 5.5; 6.1-6.3

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