Probability and Repeated Outcomes
In Probability Is a Number Between 0 and 1, we learned that a probability describes how likely an event is according to a chance model. One useful way to understand that number is through repeated outcomes: imagine repeating the same chance process under comparable conditions and keeping track of how often the event occurs.
The proportion of repetitions in which an event occurs is its relative frequency. For example, if an event occurs 18 times in 60 repetitions, its relative frequency is \(18/60=0.30\), or 30%. A probability can be interpreted as the value that this relative frequency tends to be near over a very large number of repeated trials, when the chance model and conditions stay the same.
This interpretation connects a model’s probability with observable outcomes. The probability belongs to the model; the relative frequency is calculated from repetitions that actually took place. In a finite set of trials, the relative frequency may be above or below the model probability. It does not have to match exactly.
As described in The Law of Large Numbers in Simulations, when independent trials follow the same chance model, the relative frequency tends to settle near the event’s probability as the number of trials grows. “Tends to settle near” is not a promise that every new result moves the relative frequency closer, or that a particular number of trials will produce an exact match.
What Does a 30% Chance of Rain Mean?
Suppose a forecast gives a 30% chance of measurable rain at a specified location during a specified time period. The event is rain meeting the forecast’s definition at that location and during that period. The forecast assigns probability \(0.30\) to that event.
In long-run terms, if there were many comparable forecast occasions with a 30% chance of rain, rain would occur on about 30% of those occasions. This is a way to explain what the probability means across repetitions. It is not a claim that rain must occur on exactly 30 out of every 100 occasions, and it does not tell us with certainty whether it will rain on the particular occasion we are considering.
The context matters. A 30% chance of rain does not ordinarily mean that it will rain for 30% of the day, that 30% of the forecast area will receive rain, or that rain is guaranteed somewhere within a particular area. Those are different statements about different events. Interpret the probability for the event and setting actually specified.
A single occasion has an outcome: under the stated definition, it either rains or it does not. The probability describes the chance of those possible outcomes before the result is known; it is not a fraction of a single outcome. If it rains, that one event occurred. If it does not, it did not. Either result is possible under a model that assigns a probability of \(0.30\).
Worked Example: Interpret a 30% Rain Forecast
A fictional forecast assigns a 30% chance of measurable rain at a town park between noon and 6 p.m. on Saturday. Explain the meaning of this forecast for Saturday and in the long run.
First, identify the event: measurable rain at the town park between noon and 6 p.m. The forecast assigns this event probability \(0.30\), which is 30%.
For this particular Saturday, the event may occur or may not occur. The 30% probability does not mean that it will rain for \(0.30 \times 6=1.8\) hours, nor does it guarantee that rain will stay away. Those interpretations incorrectly turn a chance for one event into a duration or a promise.
In the long-run interpretation, consider many comparable Saturday forecast occasions at that park and for that time period, each with a 30% chance of rain. The model says measurable rain would occur on about 30% of those occasions. It need not occur on exactly 30 of any particular 100 occasions. The interpretation describes a pattern across many comparable cases, not the outcome of this one Saturday.
Coin Flips: A Clear Repeated-Trial Model
A coin-flip model makes the long-run idea easy to see. Suppose a coin is modeled as fair, so the probability of heads on each flip is \(0.50\). The event of interest is “the result is heads,” and one trial is one flip. If we flip the coin a fixed number of times, we can calculate the relative frequency of heads by dividing the number of heads by the number of flips.
Here is a short illustrative sequence of 10 flips with 7 heads. Its relative frequency of heads is:
That is 70%, which is noticeably higher than the model probability of 50%. There is no contradiction. Ten flips are a small number of repetitions, and a fair-coin model allows different short-run results, including 7 heads in 10 flips. The model’s probability does not require a 5-to-5 split in every set of 10 flips.
If we repeat the experiment many more times, the relative frequency will generally tend to be near \(0.50\). It can still be a little higher or lower. There is no rule that the count of heads must equal half the number of flips, and the proportion need not move steadily closer to \(0.50\) after each added flip.
Worked Example: Relative Frequency in a Coin-Flipping Run
A student uses a fair-coin model and records 512 heads in 1,000 flips. Calculate the relative frequency of heads and compare it with the model probability.
The relative frequency is the number of heads divided by the total number of flips:
The observed relative frequency of heads is 51.2%. The model probability is \(0.50\), or 50%, so this run’s relative frequency is 1.2 percentage points above the model probability. It is close to 50%, but it is not exactly 50%.
This result is consistent with the long-run interpretation: in a large run of flips, the relative frequency can be near the probability without matching it exactly. The 512 heads do not prove that the coin is fair, and a different run could produce a different relative frequency. The conclusion here is limited to comparing this run’s observed proportion with the probability in the stated model.
Relative Frequency Is Not a Guarantee
The long-run interpretation can be misunderstood if we treat it as a guarantee for a small batch or a single trial. A probability of \(0.30\) does not require exactly 30 occurrences in every 100 trials. A probability of \(0.50\) does not require the same number of heads and tails in every group of flips. Relative frequency varies from one finite set of repetitions to another.
It is also a mistake to think that outcomes have to “make up for” earlier results. If a fair coin lands heads several times in a row, that sequence alone does not make tails certain on the next flip under the fair-coin model. The long-run pattern is about the behavior of repeated trials overall; it does not say that the next outcome must correct the running total.
The Law of Large Numbers concerns what happens as the number of comparable, independent repetitions grows. It does not say that every short run resembles the long-run proportion. Nor does it promise that a specific run will reach a particular relative frequency by a chosen number of trials. As in How Many Trials Are Enough, more repetitions generally reduce variability in a simulated relative-frequency estimate, but they do not guarantee an exact result in every run.
Worked Example: Compare Two Finite Runs
Two students each use a fair-coin model. In a fictional run of 20 flips, the first student gets 13 heads. In a separate fictional run of 200 flips, the second student gets 108 heads. Calculate both relative frequencies and explain how they relate to the model probability.
For the first student:
For the second student:
The model probability of heads is 50%. The first run’s relative frequency is 15 percentage points above the model probability; the second is 4 percentage points above it. Both runs are above 50%, and neither has to equal 50%. The larger run’s relative frequency is closer to the model probability in this example, but that does not establish a rule that every larger run will be closer than every smaller run.
These are illustrative results, not a claim that all coin-flipping runs follow this exact pattern. The example shows how to calculate and compare finite-run relative frequencies while keeping the model probability distinct from the results observed in a particular run.
Using Repeated Outcomes to Think About a Model
Repeated outcomes can help us understand a probability model. If a process is repeated under comparable conditions, we can calculate an observed relative frequency and compare it with the probability the model assigns. This is the same basic idea used in Comparing Simulated and Theoretical Probability: a simulated relative frequency estimates a theoretical probability, but a finite simulation result may differ from it.
For example, suppose a fictional package-delivery model assigns probability \(0.80\) to a package arriving by Tuesday. In a set of 50 comparable delivery occasions, 39 packages arrive by Tuesday. The observed relative frequency is:
The observed 78% is close to, but not identical to, the model’s 80%. This comparison does not prove the model is correct or incorrect. It reports what happened in those 50 cases and how that relative frequency compares with the model probability. A thoughtful interpretation also asks whether the cases really were comparable and whether the chance process stayed consistent.
The phrase “in the long run” is important because it points to many repetitions of the same kind of chance process. It does not mean “eventually the outcome has to occur,” nor does it mean that a probability applies unchanged when the setting or chance process changes. If the conditions change, the relevant model probability may change too.
Common Mistakes and AP Exam Tips
- Interpreting probability as a promise for one trial. A 30% chance does not mean rain will or will not occur on a particular day. State that both outcomes remain possible for that occasion.
- Requiring an exact percentage in every batch. A model probability of 50% does not require exactly 50 heads in every 100 flips. Say that the relative frequency tends to be near the probability over many comparable repetitions.
- Confusing an event probability with a duration or area. A 30% chance of rain is not automatically 30% of the day or 30% of a region. Name the event as the forecast defines it.
- Claiming that a larger run must be closer. The Law of Large Numbers describes a long-run tendency, not a guarantee that each added trial moves the relative frequency toward the probability.
- Mixing up probability and relative frequency. The probability is assigned by the chance model; relative frequency is calculated from observed or simulated repetitions. Identify which one you are reporting.
- Leaving out the repeated-trial context. A full-credit long-run interpretation identifies the event and says that its relative frequency would be about the stated probability across many comparable repetitions.
For an AP response, connect the probability to the event, the chance model, and the repeated-trial interpretation. Use words such as “about” or “tends to be near” rather than claiming an exact count. When discussing one trial, clearly separate what the model says from what actually happens.
Check Your Understanding
For each question, distinguish the model probability from the result of one trial or a finite run.
- A fair-coin model assigns probability \(0.50\) to heads. In 40 flips, a student gets 27 heads. Calculate the relative frequency of heads and explain whether this contradicts the model.
- A forecast gives a 20% chance of measurable snow during a specified evening. Explain its long-run interpretation for many comparable forecast occasions and state what it does not guarantee for this evening.
- In a fictional set of 80 comparable trials, an event occurs 18 times. Calculate its relative frequency as a decimal and a percentage.
- Explain why a 50% probability does not mean that an event must occur exactly 50 times in every 100 trials.
- A student says, “The coin landed heads four times in a row, so tails is due on the next flip.” Explain why this conclusion does not follow from a fair-coin model.