Chapter 4 - Basic Probability

Question 1

What is the definition of a Probability? And what number scale do all Probabilities fall on?

Answer 1

The chance that an uncertain event will occur

0-1

Question 2

What is an Impossible Event, and what number on the probability scale represents and Impossible Event?

Answer 2

An event that has no chance of occurring

0

Question 3

What is a Certain Event and what number on the probability scale represents a certain event?

Answer 3

An event that is sure to occur

1

Question 4

What are the three ways you can asses Probability?

Answer 4

A Priori - a probability based on prior knowledge. Flipping a coin.

Empirical Probability - also known as experimental probability = probabilities that are not necessarily certain but based on a sample set of information

Subjective Probability - based on experiences and opinion with no fact

Question 5

What is a Simple Event or a Simple Probability?

Give an example

Answer 5

A probability based on one single characteristic

Probability of (x) = probability of selecting a male in class

Probability of (x) = probability of selecting a Wednesday from all 365 days

Question 6

What is a Joint Event or a Joint Probability?

Give an example

Answer 6

A probability based on two or more characteristics happening at the same time

Probability of (x and y) = probability of randomly selecting a male (x) within a stats class (y)

Probability of (x and y) = probability of randomly selecting a day in January (x) and that day being a Wednesday (y) of all 365 days

Probability of (x and y) = probability of wearing jeans to class (x) and the temperature being above 70 degrees (y) during the school year

Question 7

What is a complement event, how is it denoted?

Give an example

Answer 7

A complement event is the probability of all other events occurring

It is denoted with a ‘ (prime)

Probability of (x’) = the probability of not selecting a male in class

Probability of (x’) = the probability of not selecting a Wednesday out of all 365 days

Question 8

What is a Mutually Exclusive Event?

Give an example

Answer 8

An event that cannot occur if another event does. Two events that cannot occur at the same time

Randomly choosing a day out of all 365 days.
A = choosing a day in January
B = choosing a day in February

The day chosen would not be in both months at the same time

Randomly driving down a street in Lincoln
A = 84th
B = Pioneers
You could not drive down both streets at the same time

Question 9

What is a Collectively Exhaustive Event?

Give an example

Answer 9

A collectively exhaustive event is an event within a sample set that must occur and the probabilities cover the entire sample set.

Collectively exhaustive events have a probability equal to 1

Randomly rolling a die
x = 1-3
y = 4-6.
Prob (x and y) are collectively exhaustive

Randomly selecting a day during the year
x = a weekday
y = a weekend
Prob (x and y) are collectively exhaustive

Question 10

What is a Mutually Exclusive and Collectively Exhaustive event?

Answer 10

An event that covers the entire sample space but cannot happen at the same time as the other

Randomly selecting a day during the year
x = a weekday
y = a weekend
Prob (x and y) are collectively exhaustive and mutually exclusive

Weekdays and Weekends are the only two options for days of the week, but they cannot happen at the same time.

Question 11

What is the formula for a simple probability?

Give an example

Answer 11

P(A)=(number of outcomes satisfying A )/(total number of elementary outcomes)

Probability of selecting a Wednesday out of all 365 days = 52/365

Question 12

What is the formula for a joint probability?

Give an example

Answer 12

P(A and B)=(number of outcomes satisfying A and B)/(total number of elementary outcomes)

Probability of selecting a Wednesday out of all 365 days, and that Wed. being in the month of January = 4/365

Question 13

What is the formula for a Marginal Probability?

Give an example

Answer 13

P(A)=P(A and B_1)+P(A and B_2)+⋯+P(A and B_k)

P(wed.) = P(Jan. and Wed.) + P(Not Jan. and Wed.) = 4/365 + 48/365 = 52/365

Question 14

What is the formula for a conditional probability?

Answer 14

P(A|B) = P(A and B) / P(B)
or
P(B|A) = P(A and B) / P(A)

Question 15

What is the formula for an independent probability?

Answer 15

P(A|B) = P(A)

Question 16

What is the Multiplication of probabilities formula?

Answer 16

P(A and B) = P(A|B) x P(B)
or
P(B and A) = P(B|A) x P(A)

Question 17

When using Bayes’ Theorem, what are you calculating?

Answer 17

The probability of the selected event out of all other outcomes

Question 18

Give an example for counting rule one

Answer 18

k^n

k = mutually exclusive and collectively exhaustive events
n= number of trials

Ex: rolling a fair die 3 times
6 die faces rolled 3 times
6^3 = 216 possibilities

Question 19

Give an example for counting rule two

Answer 19

(k1) x (k2) x (k3)…(Kn)

Ex: you want to go to a park (3 parks), eat at a restaurant (4 restaurants), and go see a movie (6 movies)

3 x 4 x 6 = 72 possibilities

Question 20

Give an example for counting rule three

Answer 20

n! = (n)(n-1)(n-2)….(1)

Ex: you have five books to put on a shelf

5! = 5 x 4 x 3 x 2 x 1 = 120 possibilities

Question 21

Give an example for counting rule four (permutation)

With permutations, do we care about order? If so, or not, will why? Will we have a higher or lower overall number compared to Combinations?

Answer 21

nPx= n! / (n−X)!

Ex: you have five books and only put three on the shelf

5P3! = 5! / (5−3)! = 120/2 = 60 possibilities

Question 22

Give an example for counting rule five (combinations)

With combinations, do we care about order? If so, or not, will why? Will we have a higher or lower overall number compared to Permutations?

Answer 22

nCx = n! / (X! (n−X)!

You have five books to select and you only want to read three

5! / (3! (5−3)! = 120/(6x3) = 10

Question 23

What is the formula for General Addition?

How does this formula change when the events are mutually exclusive?

Answer 23

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

When they are mutually exclusive
P(A or B) = P(A) + P(B)