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Probability Basics

What is Probability?

Probability is a way to measure how likely something is to happen. We use it every day, from checking the weather forecast to deciding if we should buy a lottery ticket. It’s a number between 0 and 1, where 0 means the event is impossible and 1 means it's absolutely certain.

A probability of 0.5 means an event has a 50/50 chance of occurring, just like a coin flip.

For situations where every outcome is equally likely, we can calculate probability with a simple formula. It's the ratio of the number of ways our desired event can happen to the total number of possible outcomes.

P(Event)=Number of favorable outcomesTotal number of possible outcomesP(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

Think about rolling a standard six-sided die. If you want to know the probability of rolling a 4, there's only one favorable outcome (rolling a 4) out of six total possible outcomes (1, 2, 3, 4, 5, or 6). So, the probability is $1/6$.

Sample Spaces and Events

To calculate probabilities accurately, we need to be clear about all the possibilities. This is where the concepts of sample spaces and events come in.

Sample Space

noun

The set of all possible outcomes of an experiment.

For rolling a single die, the sample space, often denoted by SS, is S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. It’s a complete list of everything that can happen.

Event

noun

A specific outcome or a set of outcomes from a sample space.

An event is just a piece of the sample space you're interested in. It could be a single outcome, like rolling a 5, or multiple outcomes, like rolling a number greater than 3 (the set {4, 5, 6}).

The Rules of the Game

Probability isn't arbitrary; it follows three fundamental rules, known as the axioms of probability. These axioms are the logical foundation on which everything else is built.

Rule 1: Probabilities are never negative. The probability of any event must be zero or positive. You can't have a -25% chance of rain.

P(A)≥0P(A) \ge 0

Rule 2: The probability of the entire sample space is 1. If you perform an experiment, one of the possible outcomes must occur. The chance of rolling a 1, 2, 3, 4, 5, or 6 on a die is 100%.

P(S)=1P(S) = 1

Rule 3: The probability of mutually exclusive events adds up. If two events cannot happen at the same time (like rolling a 1 and a 3 in a single roll), the probability that either one of them occurs is the sum of their individual probabilities.

For example, the event of rolling a 1 and the event of rolling a 3 are mutually exclusive. So, the probability of rolling a 1 or a 3 is:

P(1 or 3)=P(1)+P(3)=16+16=26=13P(1 \text{ or } 3) = P(1) + P(3) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}

A Few Basic Tools

With the axioms in hand, we can derive some useful rules for everyday calculations. One of the most helpful is the complement rule.

Complement

noun

The set of all outcomes in the sample space that are not in a given event.

The complement of an event A is often written as A′A' or AcA^c. Since an event either happens or it doesn't, the probability of an event happening plus the probability of it not happening must equal 1.

This gives us the complement rule:

P(Ac)=1−P(A)P(A^c) = 1 - P(A)

This rule is incredibly handy. Sometimes it's much easier to calculate the probability of an event not happening. For instance, what's the probability of rolling a die and getting a value less than 6?

You could add the probabilities for 1, 2, 3, 4, and 5. Or, you could use the complement. The only outcome you don't want is rolling a 6. Let A be the event 'rolling a 6'.

P(less than 6)=P(Ac)=1−P(A)=1−16=56P(\text{less than 6}) = P(A^c) = 1 - P(A) = 1 - \frac{1}{6} = \frac{5}{6}

Let's check your understanding of these core concepts.

Quiz Questions 1/5

Probability is a measure of:

Quiz Questions 2/5

You roll a standard six-sided die. What is the probability of rolling an even number?