Chapter 3-Probability

Question 1

Probability

Answer 1

The proportion or percentage of times the outcome would occur if we observed the random process an infinite number of times

Question 2

Law of Large Numbers

Answer 2

As more observations are collect, the proportion or rate (p-subn) of occurrences with a particular outcome converges to the probability (p) of that outcome.

Question 3

Disjoint or mutually exclusive

Answer 3

If two outcomes cannot both happen

Question 4

Addition Rule of Disjoint Outcomes

Answer 4

P(A1 or A2) = P(A1) + P(A2)
READ: The probability of outcome 1 or outcome 2 happening is equal to the probability of 1 plus the probability of 2

Where A1 and A2 represent two disjoint outcomes

This can be applied to many disjoint outcomes

Question 5

Venn Diagrams

Answer 5

Useful when outcomes can be categorized as part of or separate for two or three variables, attributes or random processes

Question 6

General Addition Rule

Answer 6

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

Where A and B are any two events, disjoint or not, and P(A and B) is the probability that both events occur

Question 7

Probability Distribution

Answer 7

A table of all disjoint outcomes and their probabilities

Question 8

Rules for Probability Distributions

Answer 8

Probability distribution is a list of possible outcomes with corresponding probabilities that satisfy three rules:
1 - The outcomes must be disjointed
2 - Each probability must be between 0 and 1
3 - The probabilities must total 1

Question 9

Sample Space

Answer 9

Set of all possible outcomes

Used to examine the scenario where an event does not occur

Question 10

Complement

Answer 10

Within the sample space, it represents all the outcomes (A-supc) that are not in a specified event (A)

P(A) + P(A-supc) = 1

P(A) = 1 - P(A-supc)

Question 11

Independent

Answer 11

When the outcome of two processes does not provide any useful info about the outcome of the other

Question 12

Multiplication Rule for Independent Processes

Answer 12

P(A & B) = P(A) x P(B)

The probability of two events from two independent processes can be calculated as the product of their separate probabilities

You can do this for many events, just keep multiplying

Question 13

Random Variable

Answer 13

A numeric quantity whose value depends on the outcome of a random event, either discrete (only integer values) or continuous (real values, including decimals).

Question 14

Expected value

Answer 14

The average outcome of a random variable

mu = E(X) = Sum of x from i=1 to k of P(X=xi)

Question 15

Sampling with Replacement

Answer 15

For each sample trial the probability is the same; draws are independent

i.e. suppose you have a bag with 5 red, 3 blue, and 2 orange chips. What is the probability of drawing a blue chip in the second draw?

Prob(2nd chip B|1st chip B) = 3/10 = .3

What is the probability of drawing a blue chip twice in a row?

Prob(1st chip B) * Prob(2nd Chip B|1st chipB = .3*.3 = .09

Question 16

Sampling Without Replacement

Answer 16

When the probability of the second sample being the same is not equal to the first sample probability, the sample has been “removed” from the population and cannot be used/counted again.

I.e., You have 5 red, 3 blue, 2 orange chips. You pull a blue chip on the first draw, what is the probability of drawing a blue chip n the second draw?

P(2nd chip B|1st chip B) = (3-1)/9 = 0.22

What is the probability of drawing two blue chips in a row?

P(1st chip B)P(2nd chip B|1st chip B) = 0.3.22 = .066