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Learn to let go. That is the key to happiness. ~Jack Kornfield. Probability. Section 4.1-4.5 Basic terms and rules Conditional probability and independence Bayes’ rule. This lady has lost 10 games in a row on this slot machine. Would you play this slot machine or another one?.
Basic terms and rules
Conditional probability and independence
Eg. Toss a die
Ans: __ out of 100 times.
** what are the random circumstances of the examples?
** what are the outcomes/sample space/event of the examples?
A probability model assigns a value to each outcome which satisfies the following properties:
** Revisit the examples: 1) build up probability models and 2) find probabilities of events
Example: randomly pick a number between 0 and 1
1. # of dots (RC: rolling a die)
2. height of a student (RC: randomly pick from the class)
Two events A and B are called mutually exclusive if the occurrence of one excludes the occurrence of the other.
The complement of an event A is the event that A does not occur, denoted as A.
Base on a survey of 1000 government employees:
# of married male employees
# of male employees
i.e. proportion of married male employees
proportion of male employees
the knowledge that B has occurred
change the probability of the occurrence of A,
i.e. P(A|B) = P(A).
Otherwise, A and B are dependent.
An event cannot be both mutually exclusive and independent (unless it is trivial i.e. probability 0):
A red die and a white die are rolled. Define the events:
A= 4 on red die; B= sum of two dice is odd.
Show that A and B are independent.
Given that P(grade A in 6204)= .60; P(grade A in 6304)= .60; P(grade A in both) =.36. Are A, B independent?
Consider a common pregnancy test
Eg. Pregnancy tests
Sensitivity = P( + | pregnancy)
False positive rate = P( + | non-preg)
Specificity = P( - | non-preg)
False negative rate = P( - | pregnancy)
Q: What is the probability that a woman with a positive result is actually NOT pregnant?
Think of the events A1, A2,…, Ak as representing all possible conditions that can produce the observable “effect” B. In this context, the probabilities P(Ai)’s are called prior probabilities. Now suppose that the effect B is observed to occur. Bayes’ theorem gives a way to calculate the probability that B was produced or caused by the particular condition Ai than by any of the other conditions. The conditional probability P(Ai|B) is called the posterior probability of Ai .
Suppose college roommates have a particularly hard time getting along with each other if they are both “Rationalists.”
A college randomly assigns roommates of the same sex.
What proportion of male roommate pairs will have this problem?
What proportion of female roommate pairs will have this problem?
Assuming that half of college roommate pairs are male and half are female.
What proportion of all roommate pairs will have this problem?
A psychologist has noticed that “Teachers” and “Rationalists” get along particularly well with each other, and she thinks they tend to marry each other. One of her colleagues disagrees and thinks that the “types” of spouses are independent of each other.