Probability Rules and Random Variables

Powerpoint Only, Not Including Web Assign

Probability is a tool to quantify

uncertainty

Probability can be interpreted as

the long run frequency of events to occur

The larger the sample size, the __ the observed outcome will be to the theoretical probability

CLOSER

Probability is a number between

0 and 1.

If the probability is 0, it is impossible

If the probability is 1, the event is guaranteed to happen.

The closer the probability is to 1, the higher the long-run frequency of the event and the more likely it is to occur. However, accurate predictions for small sample sizes still cannot be made.

Probabilities must sum to

1

What is the compliment rule?

P(A) = 1 - P(A^C)

If the probability of a customer purchasing a product after viewing an ad is 0.4, then the probability of a customer NOT purchasing, must be 0.6

A = customer purchases product after viewing ad

P(A) = 0.4

What is the addition rule?

P (A or B) = P(A) + P(B)

IF A AND B ARE DISJOINT FROM EACH OTHER (no shared outcomes)

Ex. A poll of economists found 0.2 who thought the economy was in good shape, and 0.5 who thought it was in fair shape. The probability of it being Good or Fair is 0.7.

This means the Probability of those who thought it was poor is 0.7

The General Addition Rule

P(A or B) = P(A) + P(B) - P(A INTERSECTION B)

When A and B share common outcomes, we calculate the probability of A or B by adding their separate probabilities, then subtracting the intersection.

Conditional Probability

The probability of one event (A) given that another event (B) is known to have occurred

P(A| B) = (P(A INTERSECTION B) / P(B)).

Events are said to be independent if

The probability of one event occurring has no effect on the probability of the other.

The multiplication rule says that A and B are independent if

P (A and B) = P(A) * P(B)

For continuous random variables, if the distribution of outcomes is symmetric, the ___ is a very useful probability model

Normal Distribution

Normal Distributions follow the ___ Rule, which means

68-95-99.7 Rule

68% of the values fall within 1 SD of the mean

95% of the values fall within 2 SDs of the mean

99.7% of the values fall within 3 SD’s of the mean.

Values farther than 3 SDs from the mean are outliers. (SD being standard deviations)

To get normal probability in R (LOWER TAIL)

pnorm(value, mean, SD)

To get upper probability in R

TWO OPTIONS

pnorm( value, mean, SD, lower.tail=F)

OR

1-pnorm(200,250,60)

Calculating the data of a cutoff value [ R COMMAND ]

Example : What is the value such that only 10% of months have a higher cost?

You are trying to find the cutoff value for the highest 10%

You would use qnorm(0.9,mean,SD)

and continuing, if you wanted the cutoff for the highest 5%, you would use 0.95 instead of 0.9