Basic Probability Theory: Rules & Formulas

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  • 0:03 Probability Basics
  • 0:29 Visualizing Probability
  • 1:28 Probability Rules
  • 5:13 Bayes' Theorem
  • 7:56 Lesson Summary
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Lesson Transcript
Sharon Linde

Sharon has a Masters of Science in Mathematics

Expert Contributor
Michael Lawlor

Mike has 13 years of teaching experience at the college level. He has a Ph.D. in Statistics from Purdue University. Probability and modeling are his favorite areas.

This lesson contains probability basics and rules, as well as the fundamental law of total probability and Bayes' theorem. Explore these important concepts and then see if you can answer the questions in the follow-up quiz.

Probability Basics

Probability is defined as a number between 0 and 1 representing the likelihood of an event happening. A probability of 0 indicates no chance of that event occurring, while a probability of 1 means the event will occur. If you're working on a probability problem and come up with a negative answer, or an answer greater than 1, you've made a mistake! Go back and check your work.

Visualizing Probability

There are a number of ways to visualize probabilities, but the easiest way to think about them is to use the fraction method: turn the terms into a fraction by dividing the number of desirable outcomes by the total number of possible outcomes. This will always give you a number between 0 and 1. For example, what are the chances of rolling an odd number on a 6-sided die? There are a total of six numbers and three odd numbers: 1, 3 and 5. So the probability of rolling an odd number is 3/6 or 0.5. You can use this formula when performing more difficult calculations, as we'll see later in the lesson.

In this formula:

  • P(A) is read as 'the probability of A', where A is an event we are interested in.
  • P(A|B) is read as 'the probability of A given B occurs'.
  • P(not A) is read as 'the probability of not A ', or 'the probability that A does not occur'.

Probability Rules

There are three main rules associated with basic probability: the addition rule, the multiplication rule, and the complement rule. You can think of the complement rule as the 'subtraction rule' if it helps you to remember it.

1.) The Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)

If A and B are mutually exclusive events, or those that cannot occur together, then the third term is 0, and the rule reduces to P(A or B) = P(A) + P(B). For example, you can't flip a coin and have it come up both heads and tails on one toss.

2.) The Multiplication Rule: P(A and B) = P(A) * P(B|A) or P(B) * P(A|B)

If A and B are independent events, we can reduce the formula to P(A and B) = P(A) * P(B). The term independent refers to any event whose outcome is not affected by the outcome of another event. For instance, consider the second of two coin flips, which still has a .50 (50%) probability of landing heads, regardless of what came up on the first flip. What is the probability that, during the two coin flips, you come up with tails on the first flip and heads on the second flip?

Let's perform the calculations: P = P(tails) * P(heads) = (0.5) * (0.5) = 0.25

3.) The Complement Rule: P(not A) = 1 - P(A)

Do you see why the complement rule can also be thought of as the subtraction rule? This rule builds upon the mutually exclusive nature of P(A) and P(not A). These two events can never occur together, but one of them always has to occur. Therefore P(A) + P(not A) = 1. For example, if the weatherman says there is a 0.3 chance of rain tomorrow, what are the chances of no rain?

Let's do the math: P(no rain) = 1 - P(rain) = 1 - 0.3 = 0.7

Law of Total Probability

Law of Total Probability: P(A) = P(A|B) * P(B) + P(A|not B) * P(not B)

For example, what is the probability of a person's favorite color being blue if you know the following:

  • Left-handed people have blue as a favorite color 30% of the time
  • Right-handed people like blue 40% of the time
  • Left-handed people make up 10% of the population

Let's complete the equation:

1.) P(Blue) = P(left handed) * P(like blue|left handed) + P(not left handed) * (P(like blue|not left handed)

2.) P(Blue) = (0.1)(0.3) + (0.9)(0.4)

3.) P(Blue) = .03 + .36 = 0.39

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Additional Activities

Basic Probability Rules

Part 1:

Let us consider a standard deck of playing cards. It has 52 cards which run through every combination of the 4 suits and 13 values, e.g. Ace of Spades, King of Hearts. Suppose we draw 1 card from the deck at random.

Let A = a face card (King, Queen, or Jack).

Let B = a heart.

Let C = a number (2-9 only).

Calculate the following:

a) P(A)

b) P(B)

c) P(C)

d) P(A or B)

e) P(A and B)

f) P(A | B)

g) P(not C)

h) P(A and C).


1) Are the events A and B independent or dependent?

2) What probability term applies to the events A and C?

Part 2:

Suppose a Statistics Professor has analyzed test data, and the following statements are true:

1) 95% of students who think they did well get a good grade

2) 20% of students who think they did poorly get a good grade

3) 80% of students think they did well.

Let T = the student thinks they did well and W = the student did well.

Calculate the following:

A) P(W)

B) P(T|W)


Are the events of a student doing well and them thinking they did well independent or dependent? Justify your answer mathematically.

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