A Bernoulli trial is a single random experiment with exactly two outcomes, usually called success and failure. In Intro to Probability, it’s the basic setup behind binary outcomes and the Bernoulli distribution.
A Bernoulli trial in Intro to Probability is one random experiment with exactly two possible outcomes, usually labeled success and failure. Think of one coin flip, one free-throw attempt, or one item passing a quality check, where you only care whether the result fits a chosen category.
The word "success" does not mean "good" in a moral sense. It just means the outcome you are tracking. If a problem asks for the probability of getting a head, then head is the success outcome. If the problem asks for whether a part is defective, then defective can be the success outcome if that is the event being counted.
A Bernoulli trial has a success probability p and a failure probability 1 - p. Those probabilities stay fixed for that trial setup. The trial is also treated as independent from other trials, which means one result does not change the next result. That independence is what lets probability problems stay manageable when you repeat the same kind of experiment.
A useful way to see it is through coding. Many probability problems represent success as 1 and failure as 0, which turns a Bernoulli trial into a simple random variable. Then the value of the variable tells you whether the event happened, and the probability model tells you how likely that event is.
This term is the building block for the Bernoulli distribution, and it also feeds into the binomial distribution when you count how many successes happen across several independent trials. If a problem gives you repeated yes/no outcomes, check whether each trial has only two outcomes, the same success probability, and independence. If yes, you are probably looking at a Bernoulli setup.
Bernoulli trials show up whenever Intro to Probability turns a real situation into a clean yes/no model. Once you can spot a Bernoulli setup, you can move from vague language like "something happens" to a precise probability statement with p and 1 - p.
That matters because a lot of later probability work builds on this pattern. The Bernoulli distribution describes one trial, while the binomial distribution counts how many successes happen over many repeated trials. If you mix those up, you may choose the wrong formula, the wrong sample space, or the wrong random variable.
It also trains you to read word problems carefully. The same event can be framed in different ways depending on what counts as success, so you have to identify the outcome the question is asking about before you compute anything. On homework, that often means defining the success event first, then assigning p, then finding failure as 1 - p.
Bernoulli trials are also a good checkpoint for independence. If the probability changes from trial to trial, the model breaks. That distinction shows up a lot in quizzes, especially when a problem sounds like repeated trials but actually has changing conditions or dependence between outcomes.
Keep studying Intro to Probability Unit 8
Visual cheatsheet
view galleryBernoulli distribution
A Bernoulli trial is the random experiment, while the Bernoulli distribution is the probability model that describes its outcome. If you code success as 1 and failure as 0, the Bernoulli distribution assigns probabilities to those two values. In problems, you often identify the trial first and then use the distribution to write the random variable and its probabilities.
Independent events
Bernoulli trials usually assume independence, which means one trial does not affect another. That is different from just having two outcomes. A sequence of coin flips can be modeled as independent Bernoulli trials, but a situation where the probability changes after each draw is not independent in the same way.
Binomial distribution
The binomial distribution comes from repeating Bernoulli trials a fixed number of times and counting the number of successes. If a problem asks for "how many" successes rather than the result of one trial, you may have moved from Bernoulli to binomial. The same p is used each time, and the trials are still assumed independent.
Indicator variable
An indicator variable is often how you write a Bernoulli trial in math form. It equals 1 when the success event happens and 0 when it does not. This coding makes it easy to compute expected value and connect binary outcomes to broader random-variable ideas in the course.
A quiz or problem set may give you a word problem and ask whether the setup is Bernoulli. Your job is to check three things: there are only two outcomes, the success probability stays fixed, and the trials are independent if there is more than one. Then you may be asked to name p, write the failure probability as 1 - p, or convert the situation into a Bernoulli random variable.
If the question extends to repeated trials, use the Bernoulli idea to decide whether the problem should shift into a binomial count. A common mistake is treating any yes/no situation as binomial without checking whether the trials are repeated and identical. Another common mistake is calling the physically desirable result "success" even when the problem defines success differently. Read the event first, then label it clearly.
A Bernoulli trial is one yes/no experiment. A binomial distribution counts the number of successes across several independent Bernoulli trials. If the question is about one outcome, think Bernoulli. If it asks for the number of successes in a fixed number of trials, think binomial.
A Bernoulli trial is one random experiment with exactly two outcomes, usually called success and failure.
The probability of success is p, and the probability of failure is 1 - p.
Success just means the outcome you are counting, not necessarily the outcome you personally want.
Bernoulli trials are usually treated as independent, so one trial does not change the next one.
If you repeat Bernoulli trials and count the number of successes, you are moving toward the binomial distribution.
It is a single random experiment with two possible outcomes, often labeled success and failure. In Intro to Probability, it is the basic model for binary events like yes/no, pass/fail, or head/tail when you focus on one outcome.
No. A Bernoulli trial is one trial, while a binomial distribution describes the number of successes in multiple independent Bernoulli trials. If you only have one yes/no outcome, you are still in Bernoulli territory.
Yes, depending on the problem. "Success" is just the event you choose to track, so it could be getting a defective part, missing a shot, or any other outcome the question focuses on. The label is about the model, not the value judgment.
Check whether the situation has only two outcomes, a fixed success probability, and independence from trial to trial if repeated. If any of those breaks, the problem may need a different model. Many student mistakes come from assuming every yes/no question is automatically Bernoulli.