probability - lecture 3: conditional probability and independence

What is the formula for the law of total probability?

P(A) = P(A|B1) * P(B1) + P(A|B2) * P(B2) + ... + P(A|Bn) * P(Bn)

What is the formula for joint probability?

P(A and B) = P(A|B) * P(B)

What is the formula for conditional probability?

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

What is the formula for Bayes' theorem?

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

Define the complement rule and provide an example.

The complement rule states that the probability of an event occurring is equal to one minus the probability of the event not occurring. For example, if the probability of getting a head on a coin flip is 0.6, then the probability of getting a tail is 1 - 0.6 = 0.4.

What is the law of total probability?

The law of total probability states that the probability of an event can be calculated by considering all possible ways the event can occur, taking into account the probabilities of each way.

What is Bayes' theorem?

Bayes' theorem is a formula used to calculate the probability of an event based on prior knowledge of conditions that might be related to the event.

Explain the difference between conditional probability and joint probability.

Conditional probability is the probability of an event given that another event has already occurred. Joint probability is the probability of two or more events occurring together.

Define conditional probability and give an example.

Conditional probability is the probability of an event occurring given that another event has already occurred. For example, what is the probability of getting a head on a second coin flip given that the first flip was a tail?

Explain the concept of independence in probability theory.

Independence refers to the relationship between two or more events, where the occurrence of one event does not affect the probability of the other event occurring.

How do you calculate marginal probability?

Marginal probability is the probability of a single event occurring, regardless of other events. It can be calculated by summing the joint probabilities of that event occurring with all other events.

What is the difference between mutually exclusive and independent events?

Mutually exclusive events are events that cannot occur at the same time. Independent events are events where the occurrence of one event does not affect the probability of the other event occurring.

How do you know if two events are independent?

Two events are independent if the probability of one event occurring does not affect the probability of the other event occurring.