PART 3 MODULE 3 Classical Probability, Statistical Probability, Odds

PROBABILITY

Classical or theoretical definitions:

Let S be the set of all equally likely outcomes to a random experiment.

(S is called the sample space for the experiment.)

Let E be some particular outcome or combination of outcomes to the experiment.

(E is called an event.)

The probability of E is denoted P(E).

EXAMPLE 3.3.1

Roll one die and observe the numerical result. Then S = {1, 2, 3, 4, 5, 6}.

Let E be the event that the die roll is a number greater than 4.

Then E = {5, 6}



So

EXAMPLE 3.3.2

Referring to the earlier example (from Unit 3 Module 3) concerning the National Requirer:

What is the probability that a randomly selected story will be about Elvis?



see solution





























EXAMPLE 3.3.3

An office employs seven women and five men. One employee will be randomly selected to receive a free lunch with the boss. What is the probability that the selected employee will be a woman?







see solution

























EXAMPLE 3.3.4

An office employs seven women and five men. Two employees will be randomly selected for drug screening. What is the probability that both employees will be men?





see solution































EXAMPLE 3.3.5

Roll one die and observe the numerical result. Then S = {1, 2, 3, 4, 5, 6}.

Let E be the event that the die roll is a number greater than 4.

What about the probability that E doesn't occur?

We denote this as

Then

so

Note that

This relationship (the Complements Rule) will hold for any event E:

"The probability that an event doesn't occur is 1 minus the probability that the event does occur."









EXAMPLE 3.3.6

Let F be the event that the die roll is a number less than 7.

Then F = {1, 2, 3, 4, 5, 6}

So

If an event is certain to occur, then its probability is 1.

Probabilities are never greater than 1.































EXAMPLE 3.3.7

Let G be the event that the die roll is "Elephant."

Then G = { }

So

If an event is impossible, then its probability is 0.

Probabilities are never less than 0.



We have the following scale:

For any event E in any experiment,

EXAMPLE 3.3.8

A jar contains a penny, a nickel, a dime, a quarter, and a half-dollar. Two coins are randomly selected (without replacement) and their monetary sum is determined.

1. What is the probability that their monetary sum will be 55¢?

A. 1/25
B. 1/32
C. 1/9
D. 1/10

2. What is the probability that the monetary sum will be 48¢?

A. 1/10
B. 1/9
C. 1/32
D. 0





see solution



















EXAMPLE 3.3.9

What is the probability of winning the Florida Lotto with one ticket?







see solution







Empirical or Statistical Probability

EXAMPLE 3.3.10

A carnival game requires the contestant to throw a softball at a stack of three "bottles." If the pitched softball knocks over all three bottles, the contestant wins. We want to determine the probability that a randomly selected contestant will win (event E). How can this be done? Note that the classical definition of probability does not apply in this case, because we can't break this experiment down into a set of equally likely outcomes.



For instance, one outcome of the experiment is the situation where no bottles are toppled. Another outcome is the case where 1 bottle is topples, another is the case where 2 bottles are topples, and yet another outcome is the case where all 3 bottles are toppled. However, we don't know that these outcomes are equally likely.



In cases where it is not possible or practical to analyze a probability experiment by breaking it down into equally likely outcomes, we can estimate probabilities by referring to accumulated results of repeated trials of the experiment. Such estimated probabilities are called empirical probabilities:

Empirical Probability