Finite Math Chapter 7 Test

What do you call an activity with observable results?

An experiment

What do you call the outcome of an experiment?

A sample point

What do you call the set of all possible sample points?

The sample space

What do you call a subset of a sample space?

An event

If E ∩ F = {}, then E and F are…

Mutually exclusive events that cannot occur simultaneously.

How do you describe the sample space of an experiment?

Make a probability tree and plug in these sample points into a sample space S —> S = {}

How is it useful to create a sample space?

It allows you to calculate the probability of an event occurs by picking out the sample points that match the event’s criteria.

An event E occurs M times in N trials. What is this ratio, M/N, called?

The relative frequency

What is the relative frequency called if N grows limitlessly and how is it notated?

The empirical probability of E; P(E)

What are the events {S1}, {S2},… called?

Simple events

What do you call a table of the probabilities of simple events?

A probability distribution or function P being a probability function

What does the expression 0 <= P(S_i) <= 1 mean?

The probability of the probability function can be as low as zero or as high as 1

How do you determine the empirical probability of data?

Divide the total by a certain data point to find the empirical probability of each point.

You are asked for the probability that “X“ OR “Y” happens. How do you find this probability, given the probability of each of these events?

You add the probabilities of each event because “or” implies addition.

How do you find the union of mutually exclusive events?

You add the the probabilities of each event.

If E and F are NOT mutually exclusive, how do you find P(E U F)?

P(E) + P(F) - P(E ∩ F)

If the events of two sets F and F^C are mutually exclusive, then the union S = F U F^C is a…?

Disjoint union

How do you find P(F) or P(F^C), given one or the other?

P(F) = 1 - P(F^C) or P(F^C) = 1 - P(F)

What is a uniform sample space?

Each outcome has an equal chance of occuring

If in a uniform sample space, P(E) = ?

P(E) = # outcomes in E/# outcomes in S = n(E)/n(S)

If you’re asked for the event that something happens “at most“ a certain number of times, how do you solve it with n(F)/n(S)?

Add up the probability that the event happens zero times, 1 time, two times, etc. on the top part of the equation.

If you’re asked for the probability that “at least“ a certain number occurs, how do you solve it with the n(F)/n(S) equation?

Find P(F^C) and plug this number into the equation.

Given two events of an experiment, A and B, what is the probability called that is denoted P(B | A), the probability that B will occur given A has already occured?

The conditional probability

What is the formula for conditional probability?

P (B | A) = P (A ∩ B)/P(A)

What is the equation for the product rule of probability?

P (A ∩ B) = P(A) x P(B | A)

If P (A | B) = P(A) and P(B | A) = B(B), then what are A and B in relation (or lack thereof) to each other?

Independent events

How do you check if two events are independent of each other?

Check if their intersection is equal to their product —> P(E ∩ F) = P(E) x P(F)

What is the difference between forward (a priori probabilities) and backward (a posteriori probabilities)?

Forward —> you are given a given event and find the probability of a certain result
Backward —> you are given the result and find the probability of a certain event having produced that result

Give the formula for Bayes’ Theorem

P(A | D) = P(A ∩ D)/P(D)

What is the expanded form of Bayes’ Theorem?

P(A) x P(D | A)
/
P(A ∩ D) + P(B ∩ D) + P(C ∩ D) —> P(A) x P(D | A) + P(B) x P(D | B) + P(C) x P(D | C)

P(E ∩ F^C) = ? [Think of a Venn diagram with overlapping events E and F]

P(E ∩ F^C) = P(E) - P(E ∩ F)

When do you use Bayes’ Theorem?

Given evidence/results and determining the probability of the cause/event

P event in uniform sample space?

P(E) = n(E)/n(S)

P union of mutually exclusive events

P(E U F) = P(E) + P(F)

Addition rule

P(E U F) = P(E) + P(F) - P(E intersect F)

Rule of complements

P(E^C) = 1 - P(E)

Conditional probability

P(B | A) = P(A intersect B)/P(A)

Product rule

P(A U B) = P(A) x P(B | A)

Test for independence

P(A intersect B) = P(A) x P(B)