Probability is the branch of mathematics that deals with measuring and quantifying uncertainty. Probability uses math to describe the likelihood of something happening.
We can’t know the future with certainty, but what we can do is use all the available data to make reasonable predictions based on probability.
There are two main types of probability: objective and subjective.
The basic rules of probability, like the complement rule, the addition rule, and the multiplication rule.
We’ll look at conditional probability and how to describe the relationship between dependent events.
We’ll look at Bayes’ theorem, a key formula for conditional probability and the basis for more advanced Bayesian analysis.
We’ll look at probability distributions. Probability distributions describe the likelihood of the possible outcomes of a random event and can be discrete or continuous. We’ll look at discrete probability distributions such as the binomial and Poisson
Next we’ll look at continuous probability distributions and focus on the ever-popular normal distribution.
Next we’ll also discuss how z-scores can help you better understand the relationship between data values in a standard normal distribution.
Objective and Subjective Probability
There are two main types of probability objective and subjective. Objective probability is based on statistics, experiments, and mathematical measurements. Subjective probability is based on personal feelings, experience, or judgment. Data professionals use objective probability to analyze and interpret data. There are two types of objective probability: classical and empirical.
Classical probability is based on formal reasoning about events with equally likely outcomes. To calculate classical probability for an event, you divide the number of desired outcomes by the total number of possible outcomes. There are 52 cards in a standard deck, so there is only about a 1.9% (1/52) chance of getting the ace of hearts.
Data professionals use empirical probability to describe more complex events. Empirical probability is based on experimental or historical data; it represents the likelihood of an event occurring based on the previous results of an experiment or past events. Empirical means “based on”, concerned with, or verifiable by observation or experience rather than theory or pure logic.
Data professionals use sample data to make inferences or predictions about larger populations. Inferential statistics uses probability too.
Fundamentals of Probability
The probability that an event will occur is expressed as a number between 0 and 1. If the probability of an event equals zero, there’s a no chance that the event will occur. If the probability of an event equals one, there’s a 100 percent chance that the event will occur. There are lots of possibilities in between 0 and 1. Probability measures the likelihood of random events. The result of a random event cannot be predicted with absolute certainty.
A random experiment (aka statistical experiment) is a process whose outcome cannot be predicted with certainty. All random experiments have three things in common.
- The experiment can have more than one possible outcome
- You can represent each possible outcome in advance
- The outcome of the experiment depends on chance.
To calculate the probability of a random experiment, you divide the number of desired outcomes by the total number of possible outcomes. You may recall that this is also the formula for classical probability. The probability of tossing a coin and getting heads is one chance in two. This is 1 divided 2 equals 0.5 or 50 percent. This is how to calculate the probability of a single random event.
Mutually Exclusive Events
Imagine you are tossing a coin that has a head and a tail. It is not possible to end up with both a head and a tail as a result. It is either/or. The head and tail events are mutually exclusive, meaning that if you get a head, you exclude the possibility of getting a tail.
Compliment Rule
The chance of getting any rain tomorrow is 1 minus the chance of no rain tomorrow.
Multiplication Rule
The probability of getting two tails on two consecutive cion tosses is 50% * 50%. To get a head and another head is the probability of getting one head multiplied by the probability of getting a second head.Addition Rule
The probability of getting a 2 or a 4 on a single roll of a die is 1/6 + 1/6 = 1/3 = 33.3%. Her we add the probabilities together.
Conditional Probability
The chance of passing an exam given that the student studies the day before. Conditional probability is the probability of an event occurring given that another event has already occurred. For example, if you draw a King from a deck of cards, this event changes the probability of drawing another king from that deck of cards.