Chapter 4: Probability

(True/False) The only way to use Bayes' Rule is by using the formula.

False, you can also use a tree diagram (more intuitive).

What is the complement rule?

For any event A, P(A') = 1 - P(A)

What is probability?

Given an experiment, some events are more likely to occur than others. What we want to do is assign a number for the event A that conveys the likelihood of occurrence. This is called the probability of A.

What is it called if A and B have no elements in common?

If A and B have no elements in common, they are called disjoint or mutually exclusive events. Therefore their intersection is the empty set or A ⋂ B = ∅.

What is empirical probability?

In empirical probability, the probability is determined by experimentation or from the data. Empirical probability uses the frequentist approach with a (hopefully) large sample size. In a large sample, you calculate the proportion of times that the event occurs and divide that by the total number of occurrences of the event.

What are the different types of probability?

Subjective probability and Empirical probability

What is a trial?

Every time you perform an experiment it is called a trial.

What is the notation for a conditional probability?

A conditional probability is written P(A | B) and is read the probability of A given that B occurs.

What is a random experiment?

A random experiment is an activity in which there are at least two possible outcomes and the result of the activity cannot be predicted with absolute certainty.

What is a sample space?

All of the possible outcomes. Ex. For a coin flip the sample space would be {heads, tails}

What is an event?

An event is any collection of outcomes from an experiment. The sample space is one possible event.

What is a simple event?

An event that only has one outcome.

What is an outcome?

An outcome is a result from a random experiment.

There is another way of describing statistics besides the frequentist point of view, and it's called Bayesian. What is Bayesian probability?

Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated in the light of new, relevant data (evidence).

What is the difference between disjoint events and independent events?

Disjoint has to do with the events not having any elements in common (can be represented using a venn diagram). Independence has to do with the events not affecting the probability of the other event (can be shown mathematically).

What does conditional probability mean?

It mean that if you know something about the situation, it can change the probability that something else will occur. Ex. Knowing that it is unlikely you will be struck by lightning makes you more likely to be outside in a storm, which then increases your chances of being struck by lightning.

How do you calculate P(A U B)?

P(A U B) = P(A) + P(B) - P(A ⋂ B)

How do you calculate condition probability?

P(A | B) = P(A ∩ B) / P(B)

What is the probability of A denoted by?

P(A)

What is the symbol used to describe the sample space?

S or Ω (capital omega)

What is the complement of an event A?

The complement of an event A consists of all outcomes in the sample space that are not in A. This is denoted as A'

What is subjective probability?

The definition of 'subjective' is 'based on or influenced by personal opinions.' It is an individual's personal opinion about the likelihood that a particular event will occur. Subjective probability occurs when the event can only occur once or only a few times and there is no mathematical way to calculate the probability of the event occurring. Subjective probability is normally used when predicting the odds that a particular team will win at sports or the result of an election.

What is the intersection of two sets A and B is denoted by?

The intersection of two sets A and B is denoted by an upside down U, '⋂', A ⋂ B, and consists of all outcomes that are in both A and B.

What is the union of two sets A and B?

The union of two sets A and B is denoted by a 'U' or A ⋃ B. This consists of all outcomes that are in at least one of A or B.

(True/False) For any event, A, the probability of A has to be between 0 and 1 or 0 ≤ P(A) ≤ 1

True

(True/False) The empty set has no events and is indicated by {} or ∅.

True

(True/False) Tree diagrams represent conditional probabilities.

True

(True/False) We state that an event has occurred if the resulting outcome is contained in the event.

True

(True/False) The probability of the empty set is 0 or P(∅) = 0

True, this also makes sense because if your event doesn't contain anything, it can not occur.

(True/False) Assuming that ω is an outcome of A, then P(A)=∑P(ω)

True, this is just saying that if you sum up all of the probabilities of all of the outcomes in an event, that will be the probability of the event.

(True/False) The probability of the whole sample space is one or P(Ω) = 1

True, this makes sense because the sample space is everything and so this is saying that something has to occur.

What does it mean if two events are independent?

Two events are independent if knowing that one occurs does not change the probability that the other occurs.

How can we use tree diagrams to calculate the probability of a certain path?

We can use the general multiplication rule.

When do you use Bayes' rule?

We use Bayes' Rule when you are trying to calculate a conditional probability in one direction but you only know the conditional probability in the opposite direction. You can think of this as 'Backward Bayes.' This method is NOT used when you know the intersection.

How do you calculate the probability if two or more events are equally likely to occur? (example: flipping a coin)

When each of the outcomes are equally likely to occur, then you can theoretically calculate the probability of the event.

What is the equation for probability if the probability of any outcome of any random experiment is the long term proportion of times that the outcome occurs over the total number trials in the experiment (This is called the frequentist point of view).

Where n is the number of times the outcome occurred and N is the total number of trials.​

What is the general multiplication rule?

You multiply their probabilities together.