Conditional Probability:
Refers to the change that some outcome occurs given that another event has also occurred. P(AIB), where the probability of B depends on that of A happening.
Multiplicative Rule:
The probability of an intersection of two events.
Independence regarding two events: If events A and B are independent, the probability of the intersection of A and B equals the product of the probabilities of A and B; that is,
P(A
B) = P(A)P(B)
. The converse is also true. Dependent if it does not equal the other.
Random sampling = independent:
Each member of the population has an equal, independent, and known chance of being selected.
Bayes’ Rule: Can be applied when an observed event A occurs with any one of several mutually exclusive and exhaustive events, B1, B2,…,Bk.
Quiz: For two events A and B, let P(A) = .3, P(B) =.5, and P(A
B) = .2. Use this information to answer the following questions.
-
P(AIB) is: .40
-
P(BIA) is: .67
-
A and B are independent: false
-
P(A
B) is: .6
Quiz:
In a recent survey of ethnicity and church attendance within the U.S., the percentage who report attending weekly was 29.7% among whites, 40.7% among blacks, 29.8% among Hispanics, and 26.1% among all other races combined. Given that 72% of the U.S. population identifies themselves as white, 13% as black, 6% as Hispanic, and 9% as another race, answer the following questions.
-
Let H represent Hispanics and A represent attending church weekly. What is P(AIH): .298
-
What proportion of the U.S. population is black and attends church weekly: .407 x .13 = .052
-
What percentage of Americans attend church weekly: (
29.7%
×
72%
) + (
40.7%
×
13%
) + (
29.8%
×
6%
) + (
26.1%
×
9%
) = 30.8%
-
Given an individual attends church weekly, what is the probability that person is black: P(Black I Church Attendance) = .169
-
To determine if race and church attendance are independent, one could show: P(HIA) = P(H)
Quiz:
Hypertension is a condition indicated by elevated blood pressure, and is associated with generally poor cardiovascular health. According to researchers, 24% of white American adults (i.e., men and women over the age of 18) are
hypertensive, while this is true for 32% of blacks, 21% of Hispanics, and 17% of all other racial groups combined. If 75%
of the adult population
in the U.S. is white, 12% is black, 6% is Hispanic, and 7% can be classified as belonging to another
racial group, then answer the following questions:
-
Let W represent white American adults and H represent hypertensive adults. What is P(HIW): .24
-
What is the probability that you randomly sample a non-hypertensive adult white individual from the U.S. population: 100 – 24 = 76, 76 * 74 = 57%
-
The proportion of all American adults that is hypertensive is: (
24%
×
75%
) + (
32%
×
12%
) + (
21%
×
6%
) + (
17%
×
7%
) = .243
-
Given that a randomly sampled American adult is hypertensive, what is the probability that (s)he is black: P(Black I Hypertensive) = .158
-
Race and hypertension are dependent: true
