Lecture 9- Probability

Question 1

In statistics how do we talk about/ quantify uncertainty?

Answer 1

By measuring probabilities

Question 2

What are the two types of probabilities in statistics? Which do we work with in this course?

Answer 2

Subjective: probability as the degree of belief in a statement.

Objective: probability as a measure of the relative frequency ”in the
long run” of an outcome.

We focus on objective

Question 3

What is an event in probability jargon?

Answer 3

The outcome of an experiment

Question 4

What is a sample space in probability jargon?

Answer 4

The set of all possible outcomes of an experiment

Question 5

How do we calculate the probability of a certain event happening?

Answer 5

Pr(A) = no. of experiments resulting in A/ large no. of repetitions

This is denoted by nA/N

Question 6

What are complimentary events?

Answer 6

Two events are complementary if all outcomes are either one of the two events, e.g. head or tail on fair coin.
A¯ is called the complement of A and
Pr(A) + Pr(A¯) = 1

Question 7

What are mutually exclusive events?

Answer 7

There is no intersection between the two events. In other words, events are mutually exclusive if they cannot both occur at the same time.
e.g. getting heads and tails on the same toss

Question 8

Give an example where two events are not mutually exclusive…

Answer 8

Event 1= rolling a 1
Event 2= rolling an odd number

They can both happen at the same time

Question 9

What is the mathematical rules of probability?

Answer 9

1. Probabilities take values from 0 to 1.

2. The sum of the probabilities over all possible events is 1

Question 10

What device can help us visualize probabilities?

Answer 10

Venn diagrams

Question 11

Event A= rolling a 3,4,5 or 6
Event B= rolling a 2,4 or 6

What is A compliment? What is the probability of this?

Answer 11

This is when A does not happen. This would only be true if we rolled either a 1 or 2
Pr=2/6 or 1/3

Question 12

Event A= rolling a 3,4,5 or 6
Event B= rolling a 2,4 or 6

What is the probability of A? What is the probability of B?

Answer 12

Pr(A) = 4/6 = 2/3
Pr(B) = 3/6 = 1/2

Question 13

What does U mean? What does upside down U mean?

Answer 13

If there is U it means or A U B= event a or event b

Upside down U mean and A upside down U B= event a and event b occurs

Question 14

Event A= rolling a 3,4,5 or 6
Event B= rolling a 2,4 or 6

What is the probability of A or B?
What is the probability of A and B?

Answer 14

A or B = A ∪ B = {2, 3, 4, 5, 6}, therefore Pr(A ∪ B) = 5/6

A and B = A ∩ B = {4, 6}, therefore Pr(A ∩ B) = 2/6 = 1/3

Question 15

What are conditional events? How do we write what is the probability of A given B?

Answer 15

Two events are conditional if the probability of one event changes
depending on the outcome of another event.
Pr(A | B)

Question 16

If….
A={3, 4, 5, 6}
B={2, 4, 6}

Calculate Pr(B | A) and Pr(A | B)?

Answer 16

Pr(A) = 4/6 = 2/3
Pr(B) = 3/6 = 1/2

B = {2,4,6}, therefore A | B = {4,6} and Pr(A | B) = 2/3
A = {3,4,5,6}, therefore B|A = {4,6} and Pr(B|A) = 2/4 = 1/2

Question 17

What is the multiplication rule?

Answer 17

Pr(A and B) = Pr(A ∩ B) = Pr(A) Pr(B|A)

Question 18

What is the addition rule?

Answer 18

Pr(A or B) = Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)

Question 19

How do you use the addition rule for mutually exclusive events?

Answer 19

In this case, Pr(A and B) = Pr(A ∩ B) = 0

Pr(A or B) = Pr(A ∪ B)
= Pr(A) + Pr(B) − Pr(A ∩ B)
= Pr(A) + Pr(B)

Question 20

What are independent events and how does this work for the multiplication rule?

Answer 20

Events are independent when the occurrence of one event does not
affect the outcome of another event.

Pr(B|A) = Pr(B)

Pr(A and B) = Pr(A ∩ B)
= Pr(A) Pr(B|A)
= Pr(A) Pr(B)

Question 21

Pr(A) = 2/3, Pr(B) = 1/2 and Pr(A ∩ B) = 1/3
Find Pr(B|A)

Answer 21

Pr(A ∩ B) = Pr(A) Pr(B|A)
So
Pr(B|A) = Pr(A ∩ B)/ Pr(A)
Pr(B|A) = (1/3)/ (2/3)
= 1/2

Question 22

Pr(A) = 2/3, Pr(B) = 1/2 and Pr(A ∩ B) = 1/3
Find Pr(A ∪ B).

Answer 22

Use addition rule:
Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)
Pr(A ∪ B) = 2/3 + 1/2 − 1/3
Pr(A ∪ B) = 5/6

Question 23

p(O)= 0.47

What is the probability that 3 randomly selected people have blood group
O?

Answer 23

Pr(O) × Pr(O) × Pr(O) = 0.473
= 0.104
(under the assumption of independence- if family members would not be independent events)