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2.2 · Compare classical, empirical, and subjective probability

Learn to compare classical, empirical, and subjective probability through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Probability

MATH 215 Study Topic 2.2: Compare classical, empirical, and subjective probability

Probability describes how likely an event is. An event is an outcome, or group of outcomes, that interests us, such as a spinner landing on blue. Probability is expressed as a number from

0

1

0

means the event cannot happen, and

1

means it is certain. A fraction means division, so

1/2

can also be written as

0.5

or 50%. These are different ways to express the same number. The same event may receive different probability estimates depending on the information available. This lesson compares three approaches: classical probability, empirical probability, and subjective probability. Each describes likelihood, but each relies on a different basis.

What you will learn

Define classical, empirical, and subjective probability in plain language.
Recognize the information each approach uses and when it is suitable.
Calculate an empirical probability from observed outcomes.
Compare probability values without treating different kinds of evidence as interchangeable.

1. Classical probability: count equally likely outcomes

Classical probability applies when a situation has a known, limited set of outcomes and every individual outcome is equally likely. Equally likely means that no individual outcome has a greater chance than another. For example, each face of a fair six-sided die is equally likely to appear. Fairness is an assumption about how the die behaves; having six faces does not, by itself, prove that the die is fair.

To use this approach, define the event, count the possible outcomes, and check that the individual outcomes are equally likely. Then count how many outcomes meet the event description. The number of outcomes that meet the description is the numerator; the total number of equally likely outcomes is the denominator. This is the familiar fraction rule: divide the top number by the bottom number.

For example, if two of six equally likely outcomes meet the event description, the probability is

2/6=1/3

. As a decimal, this is approximately

0.3333

; as a percentage, it is approximately 33.33%. These forms describe the same probability. The result is based on the stated model and its assumptions, rather than on recorded trials.

P(A)=\frac{\text{favourable outcomes}}{\text{equally likely outcomes}}

Use classical probability when possible outcomes are known and equally likely.
Count the event's outcomes within the full set of possible outcomes.
The result depends on the assumptions about the situation.

2. Empirical probability: use observed outcomes

Empirical probability is based on observations from actual trials. A trial is one attempt or observation under the conditions being studied. Relative frequency means the number of times an event occurred divided by the total number of trials. This fraction gives an empirical estimate of the event's probability.

For example, if a spinner lands on blue in

47

out of

100

recorded spins, the empirical probability of blue is

47/100=0.4700

, or 47%. This describes the recorded spins. It does not establish that the spinner must land on blue exactly 47% of the time in future spins.

The observations must relate to the event of interest, and the number of times the event occurred cannot exceed the total number of trials. To use an observed frequency as a guide to future chances, the trials should be carried out under conditions relevant to the future situation. If the spinner or the way it is spun changes, earlier observations may be a poor guide. A larger collection of relevant observations can provide more information, but an empirical probability remains an estimate based on data.

In an empirical study, the population is the full group or set of trials we want to understand. A sample is the smaller set of trials actually observed. The variable is the outcome recorded for each trial, such as whether a spin was blue or not blue. The population parameter is the true probability for the population of interest; the sample statistic is the observed relative frequency. The parameter may be unknown, while the statistic can be calculated from the observations.

\widehat{P}(A)=\frac{x}{n}

Empirical probability uses observed relative frequency.
The observed fraction describes the sample and estimates a probability for relevant trials.
Check that the observations and conditions match the event being discussed.

3. Subjective probability: use informed judgment

Subjective probability is a person's reasoned assessment of how likely an event is, based on available information and judgment. It can be useful when outcomes are not known to be equally likely or when there are not enough relevant repeated observations to calculate an empirical estimate. For example, a project manager might judge the chance that a delivery will arrive late using the schedule, current conditions, and experience.

A subjective probability is not simply a guess with no basis. A careful estimate identifies the event and considers relevant information. Still, two people may give different estimates because they have different information, experience, or judgment. Do not present the estimate as a measured frequency unless it actually comes from observed data.

All three approaches use the same scale, from

0

1

, so their numerical values can be compared. However, the evidence behind those values differs. Classical probability comes from a model with equally likely outcomes; empirical probability comes from recorded outcomes; subjective probability comes from informed judgment. A difference between values may reflect different assumptions or evidence rather than an arithmetic error.

Subjective probability is informed judgment, not a count of equally likely outcomes or recorded trials.
Different people can reasonably produce different subjective estimates.
Identify the basis for a probability before comparing its value with another.

4. Choose and compare the approaches

Start by asking what information the situation supplies. If it gives a complete list of equally likely outcomes, classical probability may fit. If it gives actual recorded trials, calculate empirical probability. If neither a suitable equal-outcome model nor relevant observations are available, a reasoned subjective estimate may be appropriate. A situation can be considered using more than one approach, but label each answer by its source.

For any approach, state the event in words before giving a number. Check that the probability is between

0

and

1

, inclusive. For classical probability, check that the outcomes are equally likely; listing outcomes does not make them equally likely. For empirical probability, check the event count and total trial count. For subjective probability, explain what information informs the judgment.

Probability describes likelihood, not certainty. A small probability does not mean an event is impossible, and a large probability does not guarantee that it will happen on a particular occasion. When reporting a value, give enough context for a reader to tell whether it comes from a model, observed data, or judgment. This context is essential when comparing values: identical numbers can be supported by different kinds of evidence.

Name the event, identify the evidence, and then choose the approach.
A numerical value alone does not reveal how it was obtained.
Check that probability values fall between zero and one.

Worked example

Comparing three estimates for a spinner

A spinner has two equal-sized blue sections and two equal-sized yellow sections. In a set of

100

spins, it lands on blue

47

times. A student, considering the spinner and the observed results, gives a personal estimate of

0.45

for the chance of blue on a spin. Find the classical and empirical probabilities of blue, identify the subjective estimate, and compare all three.

Define the event and check the model
Let the event be that one spin lands on blue. For the classical calculation, assume the four sections are equally likely because they have equal size and the spinner is treated as balanced. Two of the four sections are blue. This equal-likelihood condition allows the classical count.
$A=\text{landing on blue}$
Calculate the classical probability
Divide the number of blue sections by the total number of equally likely sections. These counts refer to sections, not to the number of spins previously observed.
$P(A)=\frac{2}{4}=0.5000$
Calculate the empirical probability
The $100$ recorded spins are the sample, and the recorded variable is whether each spin is blue. The sample statistic is the relative frequency: divide the $47$ observed blue outcomes by the $100$ observed spins. The result describes these spins and estimates the chance for comparable spins.
$\widehat{P}(A)=\frac{47}{100}=0.4700$
Identify and compare the subjective estimate
The student's value of $0.45$ is subjective because it is presented as a personal assessment informed by the spinner and the observations, not as a count-based calculation. The classical value is highest at $0.5000$ , the empirical value is $0.4700$ , and the subjective value is $0.4500$ . Their closeness does not make their sources identical.
$0.4500<0.4700<0.5000$

Answer: The classical probability of blue is

0.5000

, the empirical probability is

0.4700

, and the student's subjective estimate is

0.4500

. Rounded for a concise final report, these are

0.50

0.47

, and

0.45

, respectively.

Check: The classical probability uses equally sized sections as equally likely outcomes. The empirical probability uses

47

blue outcomes out of

100

trials. The subjective probability is identified by its basis in informed personal judgment.

Common mistakes and how to avoid them

Using classical probability just because the possible outcomes can be listed.

Correction: Classical probability also requires that the listed individual outcomes are equally likely.

Calling an observed proportion the exact probability for every future trial.

Correction: An empirical probability is a relative frequency from observed trials and an estimate for relevant future trials.

Treating a subjective estimate as if it were a measured frequency.

Correction: Label it as judgment and state what information informs that judgment.

Comparing numerical probabilities without noting how each was obtained.

Correction: Report whether each value is classical, empirical, or subjective, because each rests on different evidence.

Lesson summary

Classical probability counts event outcomes among known, equally likely outcomes.
Empirical probability divides observed occurrences by observed trials and describes a sample while estimating chances for comparable trials.
Subjective probability is an informed judgment and may differ between people.
To compare probability values fairly, define the event and identify the basis for each value.

Check your understanding

Question 1

A report says that

18

60

recorded deliveries arrived late. Which approach gives the value

18/60

Classical probability, because late and on-time are two possible outcomes
Empirical probability, because it uses recorded deliveries
Subjective probability, because the report discusses future deliveries
Classical probability, because the number of deliveries is known

Show answer and explanation

Empirical probability, because it uses recorded deliveries

The value comes from observed deliveries, so it is empirical. The calculation is

18/60=0.3000

, or 30% for the recorded deliveries.

Question 2

A fair coin is flipped once. What classical probability applies to landing on heads?

$0.25$ , because heads is one of four possible results
$0.50$ , because heads is one of two equally likely outcomes
$1.00$ , because the coin must land on heads or tails
$0.00$ , because a single flip is not enough data

Show answer and explanation

0.50

, because heads is one of two equally likely outcomes

A fair coin has two equally likely outcomes, and one is heads, so the classical probability is

1/2=0.50

Question 3

A supervisor estimates a

0.30

chance that a particular shipment will be delayed, based on the current schedule and experience. What kind of probability is this?

Classical, because the estimate is written as a decimal
Empirical, because it concerns a shipment
Subjective, because it is an informed judgment
Empirical, because all estimates about future events come from data

Show answer and explanation

Subjective, because it is an informed judgment

The value is based on a supervisor's assessment and is not given as a relative frequency or an equally likely outcome count.

Key terms

Event: An outcome or group of outcomes of interest, such as a spin landing on blue.
Trial: One attempt or observation in a process being studied.
Relative frequency: The number of observed occurrences of an event divided by the total number of observations.
Population: The full group or set of trials we want to understand.
Sample: The smaller group of trials or observations actually recorded.
Variable: A characteristic recorded for each observation, such as whether a spin lands on blue.
Parameter: A numerical description of a population, such as its true probability of landing on blue.
Statistic: A numerical value calculated from a sample, such as the observed fraction of blue spins.

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