Basic Concepts of Probability

Probability is the measure of the likelihood that an event will occur. It is always a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
The sample space of a probability experiment or random trial is the set containing all possible outcomes.
An event is one or more outcomes of an experiment. Probability measures are assigned to events.

Independent and Dependent Events

Independent events are those where the outcome of one event does not affect the outcome of another.
Dependent events, in contrast, are those where the outcome of one event affects the outcome of another.

Theoretical probability is calculated by dividing the number of ways an event can occur by the total possible outcomes.
Experimental probability is calculated by recording the number of times an event occurs in an experiment over a large number of trials, then dividing by the total number of trials.
The addition rule is used when the event can occur in one of two ways. The probabilities of these ways are added to find the total probability.
The multiplication rule applies when we want to find the probability of two independent events occurring together. In this case, the probabilities are multiplied.

Conditional probability is the probability of an event given that another event has already occurred.
If the events are independent, the conditional probability is simply the probability of the event. If they are dependent, it’s the probability of both events happening divided by the probability of the event you’re conditioning on.

A probability distribution is a table, graph or formula that assigns probabilities to each possible outcome.
A uniform distribution has equal probabilities for all outcomes.
Binomial distributions can be used when an experiment has two possible outcomes, like success or failure.
Like with all statistical methods, it’s important to understand the assumptions for each type of distribution and test if your data meets these assumptions before using the distribution.

Always use sound judgment when interpreting probabilities. A probability of 0.5 does not guarantee that event will occur precisely once in two trials.
Be cautious of the Gambler’s Fallacy, the mistaken belief that if something happened more frequently than normal during the past, it’s less likely to happen in the future, or vice versa. Independent events have no memory.