Introduction to Probability

KEY TERMS

• Probability: the chance or likelihood of something happening in the future

• Event: anything we can calculate the probability of

• Outcome Space: all possible outcomes of something

A probability describes the chance or likelihood of something happening in the future. Probabilities are often used to talk about events when we aren’t sure of the outcome, but we can measure how likely it is for an outcome to occur. For example, someone talking about the weather might say there is a 60% chance of rain tomorrow — this is another way to say that the probability of rain tomorrow is 60%. We won’t know for sure what tomorrow’s weather will be until tomorrow, but weather forecasters can use some information to tell you how likely it is that it will rain.

When we calculate the probability of something happening, that “something” is called an event. An event can be anything that we can calculate the probability of. For example, flipping a head when you flip a coin is an event. Flipping two heads in a row when you flip a coin twice is another event.

A probability is usually written as a number between 0 and 1, or as a percentage between 0% and 100%. A probability of 0 or 0% means there is no chance the event will happen, while a probability of 1 or 100% means the event will definitely happen. To convert between the two, you can take a probability between 0 and 1 and multiply by 100 to get a percentage. For example, if the probability that it will rain tomorrow is 0.75, there is a 75% chance it will rain tomorrow. In other words, it is more likely that it will rain tomorrow than not rain, but rain is not guaranteed!

Usually, the event we want to find the probability of is not the only possible outcome we could see. If we wanted to find the probability of flipping a coin and landing on heads, we need to consider the fact that a coin flip could land on either heads or tails. The outcome space is the term we use to describe all of the possible outcomes of something. In the case of flipping a coin, the outcome space would include two outcomes — landing on heads and landing on tails — because these are all of the possible results we could get after a coin flip. The event (landing on heads) is a part of this outcome space, but the outcome space includes other outcomes as well. This is why probabilities are useful; they can help us understand the likelihood of a desired outcome in comparison to all other possible outcomes!

In the following sections, we will be learning about different approaches that will help us calculate the probabilities of events. Probability is important in data science because we can use it to make predictions about events in the future and describe the likelihood of those events happening based on data from past observations.

Checkpoint

Question #1

What would the outcome space of rolling a six-sided die look like? In other words, what are the possible outcomes we could see after rolling a single die?

Answer

The outcome space of rolling a six-sided die would include six different possible outcomes: rolling a 1, rolling a 2, rolling a 3, rolling a 4, rolling a 5, and rolling a 6.

Question #2

What is the outcome space of flipping a coin twice? In other words, what are all of the possible results we could see after two coin flips?

Hint: The outcome after flipping a coin twice should include two parts, the result of the first flip and the result of the second flip.

Answer

The outcome space of flipping a coin twice would include four possible outcomes: {H, H}, {H, T}, {T, H}, and {T, T}. Here, each { _ , _ } represents one possible outcome of flipping a coin twice. It has two parts because the first letter represents the result of the first coin flip and the second letter represents the second flip. Notice how we consider {H, T} and {T, H} different outcomes, because we are also interested in the order.