AP Statistics 2.3 Estimating Probabilities Using Simulation Study Notes - New Syllabus

2.3.A.1 Random Process

A random process is an experiment or activity in which the result is determined entirely by chance, rather than being known in advance.

Although the possible outcomes are known before the experiment begins, it is impossible to predict with certainty which outcome will occur on any single trial.

Random processes are the foundation of probability because they produce outcomes that occur unpredictably.

Characteristics of a Random Process

• The outcome is determined by chance.
• All possible outcomes are known before the experiment begins.
• The exact outcome cannot be predicted with certainty.
• The process can usually be repeated many times under the same conditions.

Important AP Exam Notes

• A random process produces outcomes determined by chance.
• The possible outcomes are known before the experiment.
• The exact result of a single trial cannot be predicted.
• Random processes are used to model real-world probability situations.

2.3.A.2 Outcome

An outcome is the result of one trial of a random process.

• Each time a random process is performed, exactly one outcome occurs.
• An outcome is a single possible result from the experiment.

Examples of Outcomes

Important AP Exam Notes

• An outcome is the result of one trial.
• Only one outcome occurs each time the experiment is performed.
• An event may contain one or more outcomes.

2.3.A.3 Event

An event is a collection (set) of one or more outcomes from a random process.

An event occurs whenever the outcome of the random process belongs to the specified collection of outcomes.

An event may contain:

• One outcome.
• Several outcomes.
• All possible outcomes.

Examples of Events

Outcome vs. Event

Important AP Exam Notes

• An event consists of one or more outcomes.
• An event occurs if the observed outcome belongs to the specified set.
• Probability is assigned to events, which may contain multiple outcomes.

2.3.A.4 Simulation

A simulation is a method used to imitate a real-world random process by using another random process that is easier to perform.

Simulations are especially useful when calculating theoretical probabilities is difficult or when conducting the actual experiment is expensive, time-consuming, or impractical.

In a simulation, every possible outcome of the real-world event is assigned to a random mechanism (such as random digits, a spinner, a coin, or a random number generator).

The simulation is repeated many times, and the results are recorded to estimate the probability of an event.

Purpose of a Simulation

• Model a real-world random process.
• Estimate probabilities.
• Study events that are difficult or expensive to observe directly.
• Predict long-run behavior using repeated random trials.

Steps for Conducting a Simulation

1. Describe the real-world random process.
2. Assign each possible outcome to a random device (random digits, cards, spinner, etc.).
3. Perform one trial of the simulation.
4. Record the outcome.
5. Repeat the simulation many times.
6. Count how many times the event of interest occurs.
7. Estimate the probability using the relative frequency.

Simulation Probability Formula

\( \text{Estimated Probability}=\dfrac{\text{Number of Times the Event Occurred}}{\text{Total Number of Simulation Trials}} \)

Example of Assigning Random Digits

A bag contains 3 red marbles and 2 blue marbles. One marble is selected at random.

To simulate this process using random digits:

Each random digit represents one possible outcome of the real experiment.

Recording Simulation Results

Suppose the simulation is performed 20 times.

The estimated probability of selecting a red marble is

\( \dfrac{13}{20}=0.65 \)

Important AP Exam Notes

• A simulation should accurately represent the real-world random process.
• Every possible outcome must be assigned to the simulation.
• Each simulation trial should be independent of the others.
• Record both the count of the event and the total number of trials.
• The estimated probability is based on the relative frequency from the simulation.
• Increasing the number of simulation trials generally improves the accuracy of the probability estimate.

Common AP Exam Mistakes

2.3.A.5 Probability as Long-Run Relative Frequency

The probability of an outcome or event is its long-run relative frequency. This means that if a random process is repeated many times under identical conditions, the proportion of times an outcome or event occurs will approach its true probability.

Although results from a few trials may vary considerably, the relative frequency becomes more stable as the number of trials increases.

Formula

\( \text{Relative Frequency}=\dfrac{\text{Number of Times the Event Occurred}}{\text{Total Number of Trials}} \)

As the number of trials becomes very large,

\( \text{Relative Frequency}\rightarrow\text{Probability} \)

Example

A fair coin is flipped 1,000 times.

The theoretical probability of getting Heads is \(0.50\). The simulated relative frequency of \(0.507\) is very close because many trials were performed.

Important AP Exam Notes

• Probability is interpreted as a long-run relative frequency.
• Relative frequencies become better estimates as the number of trials increases.
• Small numbers of trials can produce results that differ noticeably from the true probability.

2.3.A.6 Estimating Probability Using Empirical Relative Frequency

Empirical probability is an estimate of probability obtained from observed data or repeated experiments.

Instead of using theoretical calculations, empirical probability uses the relative frequency observed in actual data.

The more observations collected, the more reliable the estimate generally becomes.

Formula

\( \text{Empirical Probability}=\dfrac{\text{Number of Times the Event Occurred}}{\text{Total Number of Observations}} \)

Example

A basketball player attempts 250 free throws and successfully makes 195.

The estimated probability of making a free throw is

\( \dfrac{195}{250}=0.78 \)

This empirical probability estimates the player’s true probability of making a free throw.

Important AP Exam Notes

• Empirical probability comes from observed data.
• It is an estimate, not necessarily the exact probability.
• Larger data sets generally produce better estimates.

2.3.A.7 Law of Large Numbers

The Law of Large Numbers states that for independent trials, as the number of trials increases, the long-run relative frequency of an outcome or event approaches its true probability.

This law explains why simulations and repeated experiments become more accurate when many trials are performed.

It does not guarantee that short-run results will match the theoretical probability.

Illustration

Notice that as the number of trials increases, the relative frequency gets closer to the true probability of 0.50.

Important AP Exam Notes

• The Law of Large Numbers applies only to independent trials.
• More trials lead to more stable relative frequencies.
• It does not mean that outcomes will alternate evenly in the short run.
• The law explains why simulations improve with larger sample sizes.
• This law does not imply that previous outcomes affect future outcomes.