The Bernoulli distribution is one of the simplest probability models in statistics, but it is also one of the most important. It describes a single trial that has exactly two possible outcomes, usually called success and failure. The probability of success is written as p, and the probability of failure is 1 - p.
This model matters because many real situations can be simplified into a yes or no outcome, such as pass or fail, defective or not defective, and heads or tails.
Understanding Statistics: The Bernoulli Distribution
The labels success and failure do not mean good and bad. They are names chosen for the outcome being counted. In a medical test, success might mean that the test detects a disease.
In a factory check, success might mean that an item is defective. Choosing the event clearly is the first important step.
The two outcomes must cover every possibility, and no result can belong to both groups. If an outcome is unclear, the data will not fit the model well.
Giving the counted outcome a value of one makes the model useful. A value of zero contributes nothing to a total, while a value of one contributes one. When many observations are added, the total becomes the number of counted outcomes.
When that total is divided by the number of trials, it becomes the observed proportion. This explains why the long run average of these zero and one values matches the chance of the counted outcome. A class survey can use this idea for responses such as owns a bicycle or does not own a bicycle.
The amount of variation depends on how balanced the two outcomes are. When the chance of success is close to zero, nearly every trial gives zero. When it is close to one, nearly every trial gives one.
Results are then fairly predictable. The greatest uncertainty occurs when each outcome has about the same chance. Coin tosses are useful examples because a fair coin is near this balanced case.
In real data, a probability near one half does not guarantee equal numbers in a small sample. It only describes the pattern expected across many similar trials.
A Bernoulli model is the building block for larger counting models. For example, each customer visit can be recorded as a purchase or no purchase. Adding the records gives the number of purchases during a period.
This step needs care. The trials should use the same success probability, and one result should not change the next result. These conditions may fail in real life.
Customers may influence one another, weather can affect attendance, and a machine may wear down over time. Students should separate the simple model from the real situation. State what counts as one trial, define success precisely, and check whether outside changes make the probability unstable.
Key Facts
- A Bernoulli random variable X has only two possible values: X = 1 for success and X = 0 for failure.
- P(X = 1) = p and P(X = 0) = 1 - p, where 0 <= p <= 1.
- The probability mass function is P(X = x) = p^x(1 - p)^(1 - x) for x = 0 or x = 1.
- The mean of a Bernoulli distribution is E(X) = p.
- The variance of a Bernoulli distribution is Var(X) = p(1 - p).
- A binomial random variable counts successes in n independent Bernoulli trials: Y = X1 + X2 + ... + Xn.
Vocabulary
- Bernoulli trial
- A Bernoulli trial is one experiment with exactly two possible outcomes, success or failure.
- Success probability
- The success probability p is the chance that the Bernoulli trial results in the outcome labeled success.
- Random variable
- A random variable is a rule that assigns a numerical value to the outcome of a random experiment.
- Expected value
- The expected value is the long-run average value of a random variable over many repetitions.
- Variance
- Variance measures how spread out the values of a random variable are around its expected value.
Common Mistakes to Avoid
- Treating p as the probability of failure is wrong because p is usually defined as the probability of success, while failure has probability 1 - p.
- Using Bernoulli for more than one trial is wrong because a Bernoulli distribution models exactly one trial, not a count across many trials.
- Forgetting that X can only be 0 or 1 is wrong because Bernoulli random variables do not take values like 2, 3, or negative numbers.
- Writing the variance as p^2 is wrong because the correct variance is p(1 - p), which depends on both success and failure probabilities.
Practice Questions
- 1 A free throw is made with probability p = 0.75. Let X = 1 if the shot is made and X = 0 if it is missed. Find P(X = 1), P(X = 0), E(X), and Var(X).
- 2 A light bulb passes inspection with probability 0.92. Model one inspected bulb as a Bernoulli random variable. What is the probability it fails inspection, and what is the variance of the random variable?
- 3 A student wants to model the number of heads in 10 coin flips using a Bernoulli distribution. Explain why this is not a Bernoulli distribution and identify the distribution that would be more appropriate.