2.3 Introduction to Probability

Probability basics

Probability is the long-run proportion of times an outcome occurs if a chance process is repeated many times (the Law of Large Numbers). Probabilities range from 0 to 1. A simulation uses random numbers or tools to imitate a chance process and estimate probabilities when theory is hard to apply directly.

Myths and misconceptions

The Gambler's Fallacy is the false belief that past outcomes affect future independent trials (e.g., thinking a coin is 'due' for tails). Short-run results can look uneven, but as trials increase, the observed proportion approaches the true probability. Independence means one outcome does not influence another.

Simulation design

A good simulation clearly states the assumed model, assigns random digits/outcomes to match probabilities, performs many repetitions, and records the outcome of interest. This lets us approximate probabilities of complex events without complicated math, and results improve as the number of trials increases.

Key terms

Probability

The long-run proportion of times an outcome occurs if a chance process is repeated many times; ranges from 0 to 1.

Law of Large Numbers

As the number of trials increases, the observed proportion of an outcome approaches the true probability.

Gambler's Fallacy

The false belief that past outcomes of independent trials affect future outcomes.

Simulation

A technique that uses random numbers or tools to imitate a chance process and estimate probabilities.

Independence (in probability)

Two events are independent if the outcome of one does not affect the probability of the other.

Try an AP-style question

Two fair dice are rolled. What is P(sum = 7)?

1. 1/11
2. 7/36
3. 1/6
4. 1/12