10.5 Probability Models, Relative Frequency, & Simulation

Probability Models, Relative Frequency, & Simulation

Section 10.1 computed theoretical and experimental probability. This section covers the standard that sits between them: build a probability model, then check it against reality. The DRC specification asks candidates to "develop a probability model and use it to find probabilities of events" and to "compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy."

What a Probability Model Is

A probability model is a complete list of the possible outcomes together with a probability for each. Two rules govern every model:

1. Each probability is between 0 and 1, inclusive.
2. All the probabilities sum to exactly 1.

If a set of numbers violates either rule, it is not a probability model — and TABE writes items that ask exactly that.

Uniform models

A uniform model gives every outcome the same probability. Rolling a fair six-sided die:

Sum: $6 \times \tfrac{1}{6} = 1$. ✓

Any event's probability is then the sum of its outcomes' probabilities: $P(\text{even}) = \tfrac{1}{6} + \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{2}$.

Non-uniform models

Most real situations are not uniform. A spinner with unequal sectors, a bag with unequal counts, or a machine with a known defect rate all need probabilities assigned individually.

A bag holds 5 red, 3 blue, and 2 green chips.

Sum: $0.5 + 0.3 + 0.2 = 1$. ✓

Finding a missing probability. A model lists $P(A) = 0.4$, $P(B) = 0.25$, $P(C) = ?$, and those are the only outcomes. Since the total must be 1, $P(C) = 1 - 0.4 - 0.25 = \mathbf{0.35}$.

Building a Model From Observed Data

When the setup is unknown — an irregular spinner, an unfamiliar machine — you build the model from relative frequency:

$$P(\text{outcome}) \approx \frac{\text{number of times it occurred}}{\text{total number of trials}}$$

A spinner is spun 200 times with these results:

The estimated model says red is nearly half the spinner's area, and green is about a tenth. Predicting forward: in 500 more spins, expect about $0.48 \times 500 = 240$ reds.

Comparing Model to Observation

TABE's version of this standard presents a theoretical model and some observed data and asks whether they agree.

A coin is flipped 40 times and lands heads 24 times. The model says $P(\text{heads}) = 0.5$, predicting 20 heads. Observed relative frequency is $24 \div 40 = 0.6$. Is the coin unfair?

Reasonable answers, in order of likelihood:

1. Ordinary random variation. Forty flips is a small sample; 24 heads is a common result for a fair coin.
2. The coin or the flipping process is biased. Possible, but 40 trials is not enough evidence.
3. A recording or counting error.

The same coin flipped 4,000 times landing heads 2,400 times is a very different matter. The relative frequency is identical at 0.6, but the sample is 100 times larger, and random variation is a far less plausible explanation. Sample size is what turns a discrepancy into evidence.

Sources of discrepancy to name

The Law of Large Numbers, Restated

As the number of trials grows, relative frequency converges toward true probability.

This is why an estimate built from 20 observations should be treated as a rough guide and one built from 2,000 as reliable — a judgment TABE asks about directly.

Simulation

A simulation models a real chance process with a simpler random device when the real situation is impractical to repeat.

Designing one takes four decisions:

1. Choose a device whose probabilities match the situation — a coin for 50/50, a die for sixths, digits 0–9 for tenths, slips in a hat for anything else.
2. Define what one trial represents.
3. Define success.
4. Run many trials and compute the relative frequency.

Problem. A store's promotion says 1 in 5 cups wins a prize. Simulate how often a customer buying 3 cups wins at least one.

• Device: digits 0–9; let 0 and 1 mean "win" (that is $\tfrac{2}{10} = \tfrac{1}{5}$) and 2–9 mean "lose."
• One trial: three digits.
• Success: at least one 0 or 1 among the three digits.
• Run 100 trials, count successes, divide by 100.

The theoretical answer is $1 - (0.8)^3 = 1 - 0.512 = 0.488$, so a well-run simulation should land near 49%. Comparing your simulated result to that theoretical value is the same model-versus-observation reasoning applied to your own experiment.