Notes on Probability | Statistics I | MTH 160, Study notes of Statistics

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Brigitte Martineau Chapter 4

Statistics I MTH

Chapter 4

Probability

Brigitte Martineau Chapter 4

In which areas of your life have you heard or used probabilities lately?

Vocabulary

 Experiment:

 Outcomes:

 Sample Space:

Notation:

Size of a sample space:

 Event: Any ___________________of the sample space

Notation:

Therefore, if A is the event of flipping a tail…

Examples:

Ex: Rolling a die

What is the sample space? S = ________________________ n(S) = ____

Let B = event of rolling an even number

B = _____________________________ n(B) = _____ P(B) = __________

Let C = event of rolling a number greater than 6

C = _____________________________ n(C) = _____ P(C) = __________

Let D = event of rolling a number less than 7

D = _____________________________ n(D) = _____ P(D) = __________

Brigitte Martineau Chapter 4

EXPERIMENT:

Let’s take a coin and flip it 10 times How many tails do you expect? ___________________

Record the number of tails: _________________________________

Trial # of tails Rel. Freq. Cum. Rel. Freq. %

Law of Large Numbers : The more times you perform an experiment the closer the experimental probability will be to the theoretical probability. The larger the number of trials, the closer the experimental probability (statistic) will be to the theoretical probability (parameter).

Number of Trials

Cum. Prob.

Brigitte Martineau Chapter 4

Example : A single die is rolled. Find the probability that the number on top is :

 A six

 An odd number

 An even number

 A number less than four

 A number less than four or even

 Not a 2

Complements

The complement of an event is the set of all sample points in the sample space that ________

belong to that event. The complement of A is denoted by ___________

Remember that P A (^ )^^ ^ P A (^ )^ ^1

Examples

 The probability that you will go out tonight is 0.6. What is the probability that you do not go out?

 The probability of your instructor canceling the next test is 0.0001. What is the probability that the test will take place next Thursday?

Characteristics of probabilities:

 P (impossible event) =

 P (event that is sure to happen) =

 A probability will always have a value between

 If you add all probabilities of all outcomes =

 P (event A) + P (not event A) =

Brigitte Martineau Chapter 4

“And” Probabilities and Conditional Probability:

What is a conditional probability?

General Multiplication Rule Let A and B be two events defined in a sample space S. Then

P A and B ( )  P A ( )  P B ( | A ) or P A and B ( )  P B ( )  P A B ( | )

Examples: In a deck of card, 2 cards are selected with replacement. Find the probability that you selected 2 face cards.

Is the probability of selecting a face on the second card affected by the fact that you already selected a face card on the first selection?

What if the previous experiment was done without replacement?

Is the probability of selecting a face on the second card affected by the fact that you already selected a face card on the first selection?

Independent Events If A and B are two independent events then

P B ( )  P B ( | A ) or P A ( )  P A B ( | )

Brigitte Martineau Chapter 4

Let’s practice: A new grading policy has been proposed by the dean of the College of Education for all education majors. All faculty and students in education were asked to give their opinion about the new grading policy. The results are shown below:

Opinion Favor (Fa) Neutral (N) Oppose (O) Row Total Students (S)^353 75 191 619 Faculty (F) 11 5 18 34 Column Total 364 80 209 653

Suppose that someone is selected at random from the College of Education (either student or faculty).

a.) Compute the P(Fa), P(Fa | F) and P(Fa | S)

b.) Compute P(F and Fa)

c.) Are the events F and Fa independent? Explain

d.) Compute P(Fa or O). Are these events mutually exclusive? Explain