Lecture 9- Probability Flashcards
In statistics how do we talk about/ quantify uncertainty?
By measuring probabilities
What are the two types of probabilities in statistics? Which do we work with in this course?
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
What is an event in probability jargon?
The outcome of an experiment
What is a sample space in probability jargon?
The set of all possible outcomes of an experiment
How do we calculate the probability of a certain event happening?
Pr(A) = no. of experiments resulting in A/ large no. of repetitions
This is denoted by nA/N
What are complimentary events?
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
What are mutually exclusive events?
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
Give an example where two events are not mutually exclusive…
Event 1= rolling a 1
Event 2= rolling an odd number
They can both happen at the same time
What is the mathematical rules of probability?
- Probabilities take values from 0 to 1.
2. The sum of the probabilities over all possible events is 1
What device can help us visualize probabilities?
Venn diagrams
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?
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
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?
Pr(A) = 4/6 = 2/3 Pr(B) = 3/6 = 1/2
What does U mean? What does upside down U mean?
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
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?
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
What are conditional events? How do we write what is the probability of A given B?
Two events are conditional if the probability of one event changes
depending on the outcome of another event.
Pr(A | B)
If….
A={3, 4, 5, 6}
B={2, 4, 6}
Calculate Pr(B | A) and Pr(A | B)?
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
What is the multiplication rule?
Pr(A and B) = Pr(A ∩ B) = Pr(A) Pr(B|A)
What is the addition rule?
Pr(A or B) = Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)
How do you use the addition rule for mutually exclusive events?
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)
What are independent events and how does this work for the multiplication rule?
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)
Pr(A) = 2/3, Pr(B) = 1/2 and Pr(A ∩ B) = 1/3 Find Pr(B|A)
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
Pr(A) = 2/3, Pr(B) = 1/2 and Pr(A ∩ B) = 1/3 Find Pr(A ∪ B).
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
p(O)= 0.47
What is the probability that 3 randomly selected people have blood group
O?
Pr(O) × Pr(O) × Pr(O) = 0.473
= 0.104
(under the assumption of independence- if family members would not be independent events)
