A Bernoulli random variable is a discrete random variable for one trial with two outcomes, usually coded 1 for success and 0 for failure. In Intro to Probability, it is the basic building block for models like the binomial distribution.
A Bernoulli random variable in Intro to Probability is the simplest discrete random variable: it describes one trial with exactly two outcomes, usually coded as 1 for success and 0 for failure. The whole point is that the random variable turns a yes/no event into a number you can work with algebraically.
The parameter is the success probability, written as p. If X is Bernoulli(p), then P(X = 1) = p and P(X = 0) = 1 - p. Nothing else can happen, so the probability mass function is just those two values. That is why Bernoulli variables show up early in the course when you first move from plain events to random variables.
Coding the outcomes as 1 and 0 is not just a convenience. It makes the mean easy to interpret: E(X) = p. So the expected value of a Bernoulli random variable is the long-run proportion of successes if you repeated the same trial many times. Its variance is p(1 - p), which is largest when p = 0.5 and smaller when the outcome is almost certain.
A common example is a quality-control check on one item, where success means the item passes inspection and failure means it does not. If the pass probability is 0.92, then X = 1 for pass and X = 0 for fail. You can immediately read the expected value as 0.92 and the variance as 0.92(0.08).
One subtle point: a Bernoulli random variable is about a single trial, not a whole collection of trials. If you repeat the same Bernoulli trial many times and count the number of successes, you move into the binomial distribution. So Bernoulli is the one-trial version, and binomial is the many-trial count version.
Bernoulli random variables are the starting point for a lot of the discrete-distribution work in Intro to Probability. Once you can model a single success/failure outcome, you can build bigger models by repeating that trial, adding outcomes, or comparing several binary variables side by side.
This term also gives you a clean way to compute expected value and variance for yes/no situations without getting lost in words. If a problem asks about a coin flip, a pass/fail test, a click/no-click event, or any other binary outcome, Bernoulli notation lets you write the setup in a compact form and then apply probability rules directly.
It also connects to joint probability work later in the course. Two Bernoulli random variables can be studied together in a joint probability table or joint PMF, which lets you analyze whether two binary outcomes move independently or not. That shows up in questions about paired measurements, two-stage decisions, or whether one event changes the chance of another.
In applications, Bernoulli models are the first stop for quality control, medical testing, and survey response coding. If you can identify the success probability and the 0/1 coding, you are halfway to the calculation the problem wants.
Keep studying Intro to Probability Unit 10
Visual cheatsheet
view galleryBinomial distribution
A binomial distribution counts how many successes happen across several independent Bernoulli trials. If one Bernoulli trial gives you 0 or 1, then a binomial random variable adds up many of those 0s and 1s. That is why Bernoulli is the one-trial building block and binomial is the repeated-trial model.
Probability mass function (PMF)
The Bernoulli random variable has a very simple PMF: one probability at 0 and one probability at 1. Writing the PMF makes the coding choice visible, so you can see exactly how p and 1 - p are assigned. This is a good warm-up for reading PMFs of larger discrete distributions.
joint probability mass function
When you study two Bernoulli variables together, you often describe them with a joint probability mass function. That lets you see the four possible pairs, like (0, 0), (0, 1), (1, 0), and (1, 1). It is the natural next step when the course moves from one binary variable to two.
success probability
The success probability p is the parameter that defines a Bernoulli random variable. Changing p changes the mean and variance, so it is the main number you identify from the story problem. Most Bernoulli questions boil down to finding the right p and matching it to the 0/1 coding.
A quiz or problem-set question will usually give you a one-trial scenario and ask you to define the random variable, find the success probability, or compute the mean and variance. Your job is to set X = 1 for success and X = 0 for failure, then use P(X = 1) = p and P(X = 0) = 1 - p. If the problem asks for expected value, you do not count outcomes by hand, you use E(X) = p. If it asks for spread, use Var(X) = p(1 - p).
You may also need to tell the difference between a Bernoulli and a binomial setup. If there is only one trial, it is Bernoulli. If the question counts successes across repeated independent trials, it has moved to binomial territory.
These are easy to mix up because both use success and failure language. A Bernoulli random variable describes one trial and takes values 0 or 1, while a binomial random variable counts the number of successes in several independent Bernoulli trials. If you see a single yes/no outcome, think Bernoulli. If you see repeated trials and a total count, think binomial.
A Bernoulli random variable is the 0/1 version of a random variable, with 1 for success and 0 for failure.
Its only parameter is the success probability p, so P(X = 1) = p and P(X = 0) = 1 - p.
The expected value of a Bernoulli random variable is p, which makes the mean easy to interpret as a long-run success rate.
The variance is p(1 - p), so the spread is largest when success and failure are equally likely.
If a problem repeats the trial many times and counts successes, you are probably in binomial distribution territory instead of Bernoulli.
It is a discrete random variable for one trial with two outcomes, usually coded as 1 for success and 0 for failure. In Intro to Probability, it is the simplest model for a yes/no event. The parameter p gives the probability of success.
If X is Bernoulli(p), then the mean is E(X) = p. That works because the variable equals 1 with probability p and 0 with probability 1 - p. The coding makes the average come out to the success probability.
Not quite. Bernoulli is one trial, so the random variable can only be 0 or 1. Binomial counts the number of successes across multiple independent Bernoulli trials. Bernoulli is the building block, and binomial is the repeated-trial count.
A pass/fail quality check is a classic example. Let X = 1 if the item passes inspection and X = 0 if it fails. If the pass probability is 0.92, then X is Bernoulli(0.92).