**A Survey of Probability Concepts**

Some Stats for Exam 2

The average grade without the extra credit was 84% - nicely done, class!

I'll bring your exams on Monday.

Probability

Here are some definitions to get you going.

Probability

- A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.

Notice that probability is a

number

between 0 and 1.

0 means that there is no chance of an event occurring.

1 means that the event will certainly occur.

Empirical Probability (aka relative Frequency)

Empirical or relative frequency is the second type of objective probability.

It is based on the number of times an event occurs as a proportion of a known number of trials.

Number of times the event occurs

Empirical probability

= -------------------------------------------

Total number of observations

Another idea that goes with empirical probability is the ...

Law of Large Numbers

- over a large number of trials, the empirical probability of an event will approach its true probability.

Mutually Exclusive versus Collectively Exhaustive

If we roll a die, we can roll any number between 1 and 6, but we can't roll

both

a 1 and a 6.

Those two events are

mutually exclusive

. If you roll a 1, then you can't roll a 6, and vice-versa.

Mutually exclusive

- the occurrence of one event means that none of the other events can occur at the same time.

So let's go over numbers 5, 8, 15, and 20.

Number five was a weighted average problem.

5. The Split-A-Rail Fence Company sells three types of fence to homeowners in suburban Seattle, Washington. Grade A costs $5.00 per running foot to install, Grade B costs $6.50 per running foot, and Grade C, the premium quality, costs $8.00 per running foot. Yesterday, Split-A-Rail installed 270 feet of Grade A, 300 feet of Grade B, and 100 feet of Grade C.

What was the mean cost per foot of fence rail installed?

a. $5.50

b. $5.75

c. $6.01

d. $6.12

e. $6.26

Grade A cost $5.00 per foot, and there were 270 feet.

So the cost of Grade A was 270 times $5.00.

Grade B cost $6.50 per foot, and there were 300 feet of grade B.

So the cost of Grade B was 300 times $6.50.

Grade C cost $8.00 per foot (geesh!), and there were only 100 feet of Grade C (I wonder why)

So the cost of Grade C was 100 times $8.00.

The mean cost is

270

($5.00)+

300

($6.50)+

100

($8.00)

------------------------------------------- = $6.12

270+300+100

Ok, now number 8.

8. What is the standard deviation of the ages of the sample of Canadian tourists?

(32, 21, 60, 47, 54, 17, 72, 55, 33, and 41)

a. 15.1

b. 17.6

c. 18.4

d. 20.4

e. None of the above

To find a standard deviation, we first have to find the average (or mean) of the numbers.

32+21+60+47+54+17+72+55+33+41

---------------------------------------------

10 (the number of observations)

432

=--------- = 43.2

10

Now we have to subtract that mean

from all of the observations.

32 - 43.2 = -11.2

21 - 43.2 = -22.2

60 - 43.2 = 16.8

47 - 43.2 = 3.8

54 - 43.2 = 10.8

17 - 43.2 = -26.2

72 - 43.2 = 28.8

55 - 43.2 = 11.8

33 - 43.2 = -10.2

41 - 43.2 = -2.2

Then we square those differences ...

125.44

492.84

282.24

14.44

116.64

686.44

829.44

139.24

104.04

4.84

And average them, but remember, this is a

sample

, so the denominator is 9 instead of 10 ...

309.76

Then take the square root

of that number ...

17.6

which is answer b.

You could have also used Excel to find the standard deviation ...

Number 15 ...

15. Find the 83rd percentile.

a. 52.5

b. 75.579

c. 85.5

d. 95.579

e. None of the above

Remember our formula on the last page?

Let's plug everything into our formula ...

which is 20 (.83) = 17.43

This means we look at the 17th number in our list, or 93.3.

We have to go .43 of the way past that. The next number is 98.6.

98.6 - 93.3 = 5.3 then 5.3 times .43 = 2.28

and 2.28 plus 93.3 is 95.58 (answer b).

Last one - number 20 ...

This answer depended on the best fit line that you or Excel drew through the data points, as well as the scale you used.

All of your answers were either 0, 2, or none of the above. If the Scantron marked your answer wrong, I added 5 points to your score.

The last bit of exam business is the extra credit.

I was explicitly looking for the term

"contingency table."

Many of you answered with "bivariate data."

Since a contingency table describes bivariate data, I gave 5 points of extra credit for both answers.

More Extra Credit Opportunities

1. Bring me your ticket stub to the Downing Planetarium show.

2. Attend the PMI dinner this Thursday. If you register now, it's free. (pmi-ccvc.org)

3. Check your Stat 7 website for an extra credit page with resources.

Now - onward and upward!

Experiment

- any process with uncertain results that can be repeated.

Outcome

- a particular result of an experiment.

Event

- a collection of one or more outcomes of an experiment.

Sample space, S

- the collection of all possible outcomes.

There are two approaches to assigning probability;

objective

and

subjective

.

Objective probability is subdivided into

1) classical probability

2) empirical probability

Let's practice the idea of

classical probability...

1. One card will be randomly selected from a standard 52 card deck. What is the probability the card will be a queen?

2. A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is white?

Number of favorable outcomes

Classical probability = --------------------------------------

Total number of possible outcomes

Exercise 1

Exercise 1 explained ...

1. There are 52 cards in the deck, or 52 possible outcomes in all.

The favorable outcome we are looking for is drawing a queen.

There are 4 queens in the deck, so ...

Number of favorable outcomes 4

Classical probability = -------------------------------------- = -------- =

.0769

Total number of possible outcomes 52

Another way to think about this is to turn this into a percent.

There is only a 7.69% chance of pulling a queen from the deck.

We could also simplify the fraction to 1/13, meaning you have a 1 in 13

of drawing a queen.

2. The jar contained 3 red marbles, 7 green marbles and 10 white marbles.

The total number of possible outcomes was 20 (or 3+7+10).

Our favorable outcome was pulling a white marble, and there were 10 of those.

Number of favorable outcomes 10

Classical probability = -------------------------------------- = -------- .5

Total number of possible outcomes 20

You have a 50% chance of pulling a white marble.

Collectively exhaustive

, on the other hand, means that at least one of the events must occur when an experiment is conducted.

For example, when you are rolling a die, you will roll either a 1, 2, 3, 4, 5, or a 6.

This set is collectively exhaustive.

Now, if the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is 1.

(More about that in another lecture - just keep that idea in the back of your brain.)

Example ...

Last semester, 80 students registered for Business Statistics 101 at Scandia University. Twelve students earned an A.

Based on this information and the empirical probability approach to assigning a probability, we estimate the likelihood of a student earning an A is .15

Empirical probability = 12 / 80 = .15

We could also convert this to a percent (15%).

More examples ...

During the 2010-2011 NBA season, Stephen Curry of the Golden State Warriors made 212 out of 227 free throw attempts.

Using the empirical approach to probability, the likelihood that he makes his next free throw attempt is .934

Empirical probability = 212 / 227 = .934

In other words, he makes the shot about 93% of the time.

On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 113 space missions for NASA.

On the basis of this information, what is the probability that a future mission is successfully completed?

Exercise 3

Exercise 3 explained ...

We are looking for the probability of a successful flight.

How many successful flights were there?

Yup, 111 successful flights (because only 2 were disasters).

There were 113 total flights.

Probability of successful flights = 111 / 113

= .98

This number is very close to one, meaning that it is

almost

certain that the flight will be successful.

Subjective Probability

If there is little or no experience or information on which to base a probability, it may be estimated

subjectively

.

Subjective concept of probability

- the likelihood of a particular event happening that is assigned by an individual based on whatever information is available.

In other words, give it your best guess.

A couple of examples of subjective probability ...

1. What is the probability that you will save one million dollars by the time you retire?

2. What is the likelihood that I will get an emu for Christmas?

Since there is not a torrent of information about these events, your best guess is as good as mine!

Approaches to Probability

Objective

Subjective

Classical Probability

Empirical Probability

Based on Available Information

Based on Equally

Likely Outcomes

Based on Relative

Frequencies

Vocab Practice!

We roll a single, fair die.

What is the experiment?

What are the outcomes?

What is the sample space (S)?

What is the probability of rolling an even number (E)?

Let's create an experiment to see how this idea of classical probability works.

Let's practice empirical probability

and the Law

of Large

Numbers.

Classical versus Objective

Suppose that a survey asked 500 families with three children to disclose the gender of their children and found that 180 families had two boys and one girl.

a. Estimate the probability of having two boys and one girl in a three child family using the empirical method.

b. Estimate the probability of having two boys and one girl in a three child family using the classical method, assuming boys and girls are equally likely.

Girolamo Cardano

Ars Magna

Liber de Ludo Aleae