Unlocking Joint and Conditional Probabilities: From Basics to the Law of Total Probability

1 Joint Probability

1.1 Definition

Joint probability is defined as the probability that two events occur together.

When discussing joint probability we implicitly assume:

A and B are independent events.

A and B occur simultaneously.

If A and B are not independent, they influence each other, so we cannot write the probability as the product of their individual probabilities.

1.2 Example

What is the probability of obtaining three heads followed by three tails when a fair coin is tossed six times?

Define the events:

A: a single toss results in heads.

B: a single toss results in tails.

Assuming each toss is independent, the joint probability can be calculated by multiplying the probabilities of the individual outcomes.

2 Conditional Probability

2.1 Definition

Conditional probability is expressed as the probability of event B occurring given that event A has occurred.

It is read as “the probability of B given A.” Event A is treated as a fact that either happens or does not happen.

2.2 Example

We roll two dice and obtain two numbers.

We want to find the probability of the following events:

A: a = 4 or b = 4

B: a + b = 7

There are two outcomes that satisfy both conditions: (3, 4) and (4, 3). Since there are 36 possible outcomes when rolling two dice, the probability can be computed accordingly.

3 Law of Total Probability

3.1 Definition

If {B_i} forms a partition of the sample space, then for any event A we have:

P(A) = Σ P(A | B_i) · P(B_i)

This formula is known as the law of total probability .

3.2 Example

A miner works under two weather conditions. The probability of completing the work on time is 0.42 when it rains and 0.90 when it does not rain. If the probability of rain is 0.45, what is the overall probability of completing the work on time?

Define the events:

R: it rains

C: the work is completed on time

The overall probability is calculated as P(C) = P(C | R)·P(R) + P(C | ¬R)·P(¬R) = 0.42·0.45 + 0.90·0.55.

4 Summary

This article introduced the concepts of joint probability and conditional probability, and built upon them to present the law of total probability, laying the groundwork for studying Bayes' theorem.