Geometric and Poisson distributions model two common types of random situations: waiting for the first success and counting rare events in a fixed interval. This cheat sheet helps students choose the correct distribution, identify parameters, and calculate probabilities quickly. It is useful for homework, test review, and interpreting real-world probability questions involving trials, rates, and event counts.
A geometric distribution uses the success probability and counts the trial number of the first success. A Poisson distribution uses the average rate and counts the number of events in a fixed time, area, distance, or other interval. The most important formulas are for geometric probability and for Poisson probability.
Key Facts
- For a geometric random variable , is the trial number of the first success, so possible values are .
- The geometric probability formula is , where is the probability of success on each independent trial.
- For a geometric distribution, the mean is and the variance is .
- The geometric cumulative probability for success on or before trial is .
- For a Poisson random variable , counts the number of events in a fixed interval, so possible values are .
- The Poisson probability formula is , where is the mean number of events in the interval.
- For a Poisson distribution, the mean and variance are both equal to , so and .
- If the rate is events per unit and the interval length is , then the Poisson parameter is .
Vocabulary
- Geometric distribution
- A probability distribution that gives the chance that the first success occurs on trial in repeated independent trials.
- Poisson distribution
- A probability distribution that gives the chance of observing events in a fixed interval when events occur at an average rate .
- Success probability
- The value is the probability that one trial results in a success.
- Rate parameter
- The value is the average number of events expected in the chosen interval.
- Expected value
- The expected value is the long-run average value of the random variable .
- Independence
- Independence means the outcome of one trial or event does not change the probability of another.
Common Mistakes to Avoid
- Using for a geometric probability is wrong because there are only failures before the first success, so the exponent must be .
- Starting a geometric distribution at is wrong for the trial-count version because the first possible success occurs on trial .
- Using the Poisson formula when events do not occur independently is wrong because the model assumes one event does not make another event more or less likely.
- Forgetting to adjust to match the interval is wrong because must describe the same time, area, or distance used in the question.
- Confusing with is wrong because a cumulative probability adds several outcomes, while an exact probability uses only one value of .
Practice Questions
- 1 A basketball player makes a free throw with probability . What is the probability that the player makes the first free throw on attempt ?
- 2 A website gets an average of sign-ups per hour. Using a Poisson model, what is the probability of exactly sign-ups in one hour?
- 3 A machine has an average of defects per meter of fabric. What is the probability of exactly defects in meters of fabric?
- 4 A student wants to model the number of cars passing a checkpoint in minutes. Explain why a Poisson distribution may be appropriate, and state one condition that should be checked.
Understanding Geometric & Poisson Distributions
The conditions behind a model matter more than the formula. For a geometric model, each attempt must have only two outcomes that can be called success or failure. The chance of success must stay the same from one attempt to the next.
Attempts must be independent, meaning an earlier result does not change a later chance. Tossing a fair coin until the first head is a good fit. Drawing cards without replacement is usually not, since each card drawn changes what remains.
Students should state what counts as success before doing any calculation. Success does not have to mean something good. It could mean finding a defective item or getting a busy signal.
Geometric problems often test careful reading of phrases. The probability of the first success on trial four means three failures occur before the success. At least four trials means the first three trials were failures.
More than four trials means the first four trials were failures. These events use slightly different numbers of failures. A useful check is that a larger value of the success probability should make long waits less likely.
The distribution has a long right tail because there is no fixed maximum waiting time. A few very long waits can pull the mean upward, especially when success is unlikely.
A Poisson model needs events to occur separately and at a roughly steady average rate. Events should not arrive in clusters caused by one another. For example, the number of misprints on a long roll of paper may be modeled this way when defects occur independently across the material.
The number of customers entering a shop may fail the model during a sale, because arrivals can surge together. The interval must be chosen clearly.
If an average is given per hour but the question asks about fifteen minutes, the rate must be scaled to the shorter interval before finding probabilities. Units are part of the calculation, not extra decoration.
Poisson results have patterns that help students judge whether an answer is sensible. When the average count is small, zero events can be quite likely and the distribution is strongly skewed to the right. As the average count grows, the shape becomes more balanced, though it is still a count distribution and cannot produce fractions or negative values.
Since the mean and variance are equal in an ideal Poisson model, real data with much more spread than its average may show clustering or changing conditions. In applications such as calls to a help desk, flaws in manufactured parts, radioactive emissions, or traffic accidents, a model is an approximation. Data, context, and units should always be checked before trusting a numerical result.