The Basics of Probability

What are we talking about when we talk about probability? In general, probability is a number that reflects the likelihood that a certain event will occur. All probabilities can be quantified from 0 to 1 (or, alternatively, from 0% to 100%), inclusive. The closer the probability is to zero, the lesser the likelihood the event has of occurring; the closer a probability is to 1, the greater the likelihood the event will occur.


Visualizing probability. There is a greater probability/higher likelihood of finding a black dot in the center of the image than there is of finding a dot towards the edges of the image. Photo by Richard Parsons.

Even those of us without backgrounds in math or statistics are somewhat familiar with probabilities and how to use them. For example, if you look at the weather report for the day and see that there’s a “90% chance of rain,” you know that you should probably bring an umbrella. However, a “10% chance of rain” might prompt you to leave the umbrella at home. But where do these numbers come from? Let’s first look at two extreme cases—certain events and impossible events—and then look more in depth at how to calculate theoretical probabilities.

Basics of Probability: Certain events and impossible events

If an event has a probability of 1, or 100%, it is a certain event. For example, in a coin toss, the probability that the coin lands either heads or tails is 100%. These are the only possible outcomes, and it’s certain that one of them will occur.

By contrast, an impossible event has a probability of 0, or 0%. An example would the probability of drawing five kings from a fair, standard deck of 52 cards. The reason this event is impossible is because there are only four kings in the deck.

Are the following events certain events, impossible events, or neither?

• The probability that a golden retriever is a dog
• The probability that a dog is a golden retriever
• The probability that Tuesday falls on a weekend
• The probability that it rains today
• The probability that placing an infinite number of bets at the roulette table in Las Vegas will cause you to go broke

(certain, neither, impossible, neither, certain)

Calculating theoretical probability

Consider the following bag and the eight colored marbles, all equal in size and weight, that it contains.


A “bag of marbles”, gorgeously rendered by this author using Microsoft Excel.

If a random marble is drawn from the bag, what is the probability that it is the black marble? We could find this by applying the following formula:


Our desired outcome is what we want. Therefore, if we want to pick the black marble, then we have only one desired outcome out of a total of eight total possible outcomes (eight total marbles). Therefore the probability that we pick the black marble is ⅛, or 0.125, or 12.5%. We often write this as follows:

P(black) = ⅛

where the notation P(…) indicates the probability of the event in parentheses.

Probability and sample spaces

The bag and the marbles it contains can be considered a sample space. Technically speaking, a sample space contains all the values that a random variable can take. To translate that away from math-speak, a sample space contains all of our possible outcomes. In the sample space above, all of the outcomes are equally likely. This will not always be the case, but for this introductory article on probability, we will only consider sample spaces that contain equally likely outcomes.

To summarize, a theoretical probability can be calculated by dividing the number of desired outcomes by the total number of possible outcomes as defined by the sample space.

For more on sample spaces and probability basics, check out this page from Boston University.

Basics of probability: “Or”

Now let’s find the probability that we draw a black OR red marble from the bag above. This increases our number of desired outcomes, since now either black or red will do. Since there is one black marble and two red marbles, our total number of desired outcomes is now three. The total number of possible outcomes is unchanged (the number of total marbles is constant), so:

P(black or red) = ⅜

Note that this is the same as the sum of the probability of picking a black marble and the probability of picking a red marble:

P(black) + P(red) = ⅛ + 2/8 = ⅜

We can summarize this rule as follows:

P(A or B) = P(A) + P(B)

The rule of thumb here is that OR in probability often involves adding probabilities together.

Basics of probability: “AND”

What is the probability of picking two red marbles in a row? So far, we know how to find the probability of picking the first red marble:

P(red) = 2/8 = ¼.

Now, given that we’ve already picked that first red marble and it’s been removed from the bag, what’s the probability we reach in again and pick the other red marble? Now, both our desired outcomes and our total number of outcomes have both decreased by one. Why? For one, because now there is only one red marble left in the bag, so there is only one desired outcome. Also, because the first red marble has been removed, we now only have 7 total marbles left in the bag. Therefore:

P(2nd red) = 1/7

Now that we have both of these probabilities, what do we do with them? If we want to know the probability of drawing a red AND THEN another red, we simply multiply the probabilities:

P(red followed by a red) = ¼ × 1/7 = 1/28

As we might expect, the probability of drawing two reds in a row is less than drawing either red separately. We can summarize this rule as follows:

P(A and B) = P(A)× P(B | A)

The notation P(B | A) can be read “the probability of B given that A has already occurred.” In this context, it would be the probability of drawing the second red marble given that the first red marble has already been drawn from the bag. Probabilities with a “given that” in them are called conditional probabilities, since they rely upon some condition that has already occurred. For more on conditional probabilities, check out this article.

More probability practice

Now that we’re clear on some definitions, let’s do some more practice. What is the probability…

• Of drawing two green marbles in a row?
• Of drawing a red or a blue marble?
• Of drawing a white marble?
• Of NOT drawing a blue marble?
• Of drawing a red, blue, orange, black, or green marble?
• Of drawing a red, blue, and black marble, in any order?

(1/28, ½, 0, ¾, 1, 1/14)

Still have questions on probability basics? Check out our statistics blog and videos!

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