Chapter 4 Joint, Marginal, and Conditional Probability

Question 1

What is the definition of joint, marginal and conditional probability?

Answer 1

ˆ Joint Probability: Probability of events A and B, The probability of two (or more) events is called the joint probability.

ˆ Marginal Probability: the probability of an event irrespective of the outcome of other variables. Probability of event A given variable Y (The probability of one event in the presence of all (or a subset of) outcomes of the other
random variable is called the marginal probability or the marginal distribution).

ˆ Conditional Probability: Probability of event A given event B.

Question 2

How is joint probability shown? P 39

Answer 2

P(A and B) = P(A ∩ B) = P(A, B)

Question 3

How is joint probability calculated? P 40

Answer 3

The joint probability for events A and B is calculated as the probability of event A given event B multiplied by the probability of event B. This can be stated formally as follows:
P(A ∩ B) = P(A given B) × P(B)

Question 4

How is marginal probability calculated? P 40

Answer 4

It is the sum or union over all the probabilities of all events for the second variable for a given fixed event for the first variable.
P(X = A) = sum (P(X = A, Y = y) | y∈Y)

Question 5

How is conditional probability shown? P 40

Answer 5

P(A given B) = P(A|B)

Question 6

How is conditional probability calculated? P 41

Answer 6

The conditional probability for events A given event B is calculated as follows:
P(A|B) = P(A ∩ B)/ P(B)

Question 7

Does the notion of A given B, means that the occurrence of B is certain? Explain your answer P 41

Answer 7

The notion of event A given event B does not mean that event B has occurred (e.g. is certain); instead, it is the probability of event A occurring after or in the presence of event B for a given trial.

Question 8

How is joint,marginal and conditional probability calculated for independent events? P 41

Answer 8

Joint Probability : P(A ∩ B) = P(A) × P(B)
Marginal Probability : P(A)
Conditional Probability : P(A|B) = P(A)
We refer to the marginal probability of an independent probability as simply the probability.
Similarly, the conditional probability of A given B when the variables are independent is simply
the probability of A as the probability of B has no effect.

Question 9

What is exclusivity? P 42

Answer 9

If the occurrence of one event excludes the occurrence of other events, then the events are said to be mutually exclusive. If the probability of event A is mutually exclusive with event B, then the joint probability of event A and event B is zero.

Question 10

For exclusive events the probability of an outcome can be described as event A or event B, how is it stated? how is it calculated? P 42

Answer 10

The or is also called a union and is denoted as a capital U letter:
P(A or B) = P(A ∪ B)
It’s calculated as below:
P(A or B) = P(A) + P(B)

Question 11

How is the union of events calculated for non-mutually exclusive events? P 42

Answer 11

The probability of non-mutually exclusive events is calculated as the probability of event A and
the probability of event B minus the probability of both events occurring simultaneously. This
can be stated formally as follows:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Question 12

Is P(A|B) = P(B|A) true?

Answer 12

No

Question 13

Explain the reason why Conditional probability is calculated like this:
P(A|B) = P(A ∩ B)/ P(B)
External-source Chapter 5 P 50

Answer 13

Probability of A happening, given B, means the probability of event A occurring after or in the presence of event B for a given trial.
In the weather example, let’s say we want the probability of sunny weather in city 1 given Sunny weather in city 2, and the weather in the two cities are DEPENDENT:
the dependence means that when one of the possible events in sample space city 2 happens, there’s a group of events in sample space city 1 that also happen, and each group contains different probabilities for each event of the sample space 1. That’s why we use P(A ∩ B)/ P(B) to calculate P(A|B). Division by P(B) can be translated as: since we are calculating the probability in a sub-sample space (where B happens in sample space 2), then we need to take into account the probability of choosing that sub-sample space in a given trial, hence the division.
Table for city1 and city2:
City 1
Sunny Cloudy Rainy Marginal Probability
Sunny 6/20 2/20 0/20 8/20
City 2 Cloudy 1/20 5/20 2/20 8/20
Rainy 0/20 1/20 3/20 4/20

So to calculate P(C1=Sunny | C2=Sunny)= (6/20) / (8/20)