The negative binomial distribution models repeated independent Bernoulli trials until a fixed number of successes occurs. This cheat sheet helps students recognize the setup, choose the correct parameterization, and compute probabilities accurately. It is especially useful in college statistics courses because the distribution appears in reliability, quality control, biology, and repeated sampling problems.
The two most common forms count either the total number of trials needed to get successes or the number of failures before the th success. The probability mass function uses a binomial coefficient to count possible trial sequences and a power of for successes. The mean and variance depend strongly on the success probability , so interpreting the parameters is as important as substituting into formulas.
Key Facts
- If is the total number of trials needed to get successes with success probability , then has support .
- For the trials-counting form, the PMF is for .
- If is the number of failures before the th success, then and the support is .
- For the failures-counting form, the PMF is for .
- For equal to total trials, the mean is and the variance is .
- For equal to failures before successes, the mean is and the variance is .
- When , the negative binomial distribution reduces to the geometric distribution, with for .
- The trials must be independent, each trial must have the same success probability , and the process must stop when the th success occurs.
Vocabulary
- Bernoulli trial
- A Bernoulli trial is a random trial with exactly two outcomes, usually called success and failure, where the success probability is .
- Success probability
- The success probability is the fixed probability that one independent trial results in a success.
- Negative binomial distribution
- A negative binomial distribution gives probabilities for the number of trials or failures needed to obtain a fixed number of successes.
- Support
- The support is the set of possible values a random variable can take, such as for total trials.
- Probability mass function
- A probability mass function assigns a probability to each possible value of a discrete random variable.
- Parameterization
- A parameterization is the chosen way to define the random variable, such as counting total trials or failures before the th success.
Common Mistakes to Avoid
- Using the wrong support, such as allowing , is wrong because at least trials are required to get successes.
- Confusing total trials with failures is wrong because and differ by , so and their means are different.
- Using instead of in the trials-counting PMF is wrong because the last trial must be the th success.
- Swapping and is wrong because the exponent on counts successes and the exponent on counts failures.
- Applying the negative binomial model when trials are not independent or changes is wrong because the standard PMF assumes identical independent Bernoulli trials.
Practice Questions
- 1 Let be the total number of trials needed to get successes when . Find .
- 2 Let be the number of failures before the th success when . Find and .
- 3 A basketball player makes each free throw independently with probability . What is the probability that the player makes the th successful free throw on the th attempt?
- 4 Explain why the last trial must be a success in the PMF when counts total trials until the th success.
Understanding Negative Binomial Distribution Reference
The combination term comes from the fact that the last trial is not flexible. If a process ends on the eighth trial with its third success, trial eight must be a success. Among the first seven trials, there must be exactly two successes and five failures.
The combination counts every possible placement of those two earlier successes. Each valid sequence has the same probability because it contains the same number of successes and failures. This is why the distribution has a combination term, a success probability raised once for each required success, and a failure probability raised once for each earlier failure.
A common source of errors is mixing up what the random variable counts. In one version, a result of eight means eight attempts were needed. In the other version, a result of five means five unsuccessful attempts occurred before the final required success.
These describe the same event in different units. The difference is the fixed number of successes being waited for.
Before using a table, calculator, or software command, write a short sentence defining the count. This step prevents off by one errors, especially when the required number of successes is one.
The model can be understood as adding several waiting times. Waiting for three successes means waiting for the first success, then the next one, then the next one. Each piece behaves like a geometric waiting time when the conditions remain unchanged.
This explains why lower success probabilities produce longer and more variable waits. It also shows why rare successes create a long right tail.
A few runs may take far more trials than expected. The mean is useful for planning, but it does not mean most runs finish exactly near that value.
Check the assumptions against the real process before applying the model. Repeated manufacturing tests may be close to independent if each item is made under stable conditions. Repeated shots by a player may not be independent because fatigue, practice, or changing defense affects later attempts.
Sampling without replacement from a small group changes the chance of success after each draw. In these cases, a negative binomial calculation can look precise while describing the wrong situation.
For probability questions involving at least or more than a stated number, it is often easier to use a cumulative probability and subtract from one. Pay close attention to whether the endpoint includes the final success, since that wording determines which count is being used.