BasiC Probability
Review (Normal Distribution)
Review (Normal Distribution)
Z-score
- Way to standardize values.
z=
𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡−𝑚𝑒𝑎𝑛
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Review (Normal Distribution)
Basic Concepts in Probability
Sample space (𝛺)
-
All the possible outcomes
Event
-
Subset of a sample space
Basic Concepts in Probability
Assume that you’re collecting data on the religious affiliation of
UST students.
A possible event would be that some of them are catholic.
Let C = {Catholic}
A possible sample space would be as follows: 𝛺 = {C, M, B}
Basic Concepts in Probability
In cases where all outcomes are equally likely:
P(A) =
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝐴
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝛺
Basic Concepts in Probability
Example:
What is the probability of drawing a king in a deck of cards?
Basic Concepts in Probability
Example:
What is the probability of drawing a king in a deck of cards?
# 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝐴
P(A) = # 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝛺
How many kings are in a deck of cards?
How many cards are there in total?
Basic Concepts in Probability
Example:
What is the probability of drawing a king in a deck of cards?
How many kings are in a deck of cards?4
How many cards are there in total? 52
P(K) =1/13
Axioms in Probability
- P(A) is always greater than or equal to 0.
- The sum of the outcomes equals one.
- If A and B are mutually exclusive, P(A or B) = P(A) + P(B)
Conditional Probability
Conditional Probability
- The idea is you condition your analysis of P(A) on a certain B
that has occurred.
Ex.
Probability of smoking if one has smoker friends
Probability of agreeing to Pura Luka Vega’s punishment if one
attends church or not
Conditional Probability
Definition
-
Probability of an event given the occurrence of another event.
If we know B has occurred, what is the probability of A?
P(A|B) =
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐵)
Note: P(A|B) does not, typically equal to P(B|A)
Conditional Probability
Which has a higher probability of occurring?
P(Intelligent|UPD Socio)
is high
P(UPD Socio|Intelligent)
is low
Conditional Probability
Assume that the contingency table below shows hypothetical
data on one’s sex and fiscal policy views.
Conservative
Liberal
Men
30
40
Women
10
50
What is the probability of choosing a man?
What is the probability of choosing a conservative who is a
woman?
What is the probability that a randomly selected man is a liberal?
Conditional Probability
Assume that the contingency table below shows hypothetical
data on one’s sex and one’s fiscal policy views.
Conservative Liberal
Total
Men
30
40
70
Women
10
50
60
Total
40
90
130
What is the probability of choosing a man?
P(M) =
70
130
= 0.54
Conditional Probability
Assume that the contingency table below shows hypothetical
data on one’s sex and one’s fiscal policy views.
Conservative Liberal
Total
Man
30
40
70
Female
10
50
60
Total
40
90
130
What is the probability of choosing a conservative who is a
woman?
P (C and W) =
10
130
= 0.077
Conditional Probability
Assume that the contingency table below shows hypothetical
data on one’s sex and one’s fiscal policy views.
Conservative Liberal
Total
Men
30
40
70
Women
10
50
60
Total
40
90
130
What is the probability that a randomly selected man is a liberal?
Conditional Probability
Conservative Liberal
Total
Male
30
40
70
Female
10
50
60
Total
40
90
130
What is the probability that a randomly selected man is a liberal?
𝑃 (𝐿 𝑎𝑛𝑑 𝑀)
P(L|M) =
𝑀
40/130
40
=
= = 0.57
70/130
70
Independent Events
If A and B are independent, then P(A|B) = P(A)
“Two events are independent if the probability associated with the
first event is not altered when the second event is known”
(Wilcox, 2017, p. 59).
Practice Exercise
Assume that the data below shows hypothetical values for
religiosity (1-religion, 0-nonreligious) among two colleges
1
0
Total
CSSP
25
45
70
CHE
35
30
65
Total
60
75
135
What is the P(1)=?
What is the P(1|CSSP) =?
Practice Exercise
Assume that the data below shows hypothetical values for
religiosity (1-religious, 0-nonreligious) among two colleges
1
0
Total
CSSP
25
45
70
CHE
35
30
65
Total
60
75
135
What is the P(1) = 60/135 = 0.44
What is the P(1|CSSP) = 0.36
Reference
Wilcox, R. (2017). Modern statistics for the social and behavioral
sciences: A practical introduction. CRC press.