Notes on Empirical Probability & Probability Rules
Empirical Probability
What is empirical probability?
Used when we don’t know enough about an experiment to calculate theoretical probabilities.
Defined by observing outcomes and using those observations to estimate probability.
The probability of an event is proportional to how many times we observe it occurring.
Empirical probabilities are also called relative frequencies.
Empirical Probability Formula
P_E = \frac{\text{# of favorable outcomes for } E}{\text{Total # of outcomes in the sample space}}
These probabilities are derived from observed data (frequencies) rather than a theoretical model.
Relative frequency: the numerator (favorable observations for E) divided by the total number of observations.
Uniform Probability Formula (reminder)
P_E = \frac{\text{# favorable outcomes for } E}{\text{Total # of outcomes in the sample space}}
When all outcomes are equally likely, this matches the counting-based approach.
Connection between empirical and theoretical probabilities
Empirical: probabilities are relative frequencies obtained from measurements.
Theoretical: probabilities are calculated from a model (which can incorporate data).
You can use the same observation data to compute either type:
Empirical: from observed frequencies.
Theoretical: from a model but can be informed by data.
Example: goldfinch probability
Data from 47 observed birds: 8 are goldfinches, with totals 15 scrub-jays, 17 towhees, 8 goldfinches, 7 hummingbirds.
Empirical probability: P_{goldfinch} = \frac{8}{15 + 17 + 8 + 7} = \frac{8}{47} \approx 0.17
Theoretical probability would use a model, but calculation can resemble the empirical form if the model aligns with observed proportions.
Mindfulness Research Project
Impact area
Investigates impact of mindfulness on procrastination, stress, and sense of belonging among students.
Context: procrastination identified as a major barrier to academic success; pandemic effects may have reduced sense of belonging.
Hypothesis
Mindfulness practices can reduce procrastination and anxiety, improve academic success, and enhance sense of belonging.
Intervention plan
Weekly mindfulness practice in class.
Activities focused on reducing procrastination.
Evaluation design
Pre-project survey and post-project survey (in-class and online).
Participation is voluntary and anonymous; if a student took the survey for another class, they should not retake it here.
Scheduling note
The survey will capture baseline and post-intervention measures to assess impact.
Additional context
Introduction to the Mindfulness research project is presented in class materials (Mindfulness Project).
Probability Rules
Overview
In addition to the empirical and uniform probability formulas, there are rules to compute probabilities indirectly.
Key rules discussed: Addition Rule (for mutually exclusive events) and Non-occurrence Rule (Complement Rule).
These rules apply to empirical probabilities but are true for any probability calculation.
Mutually exclusive events
Definition: E and F are mutually exclusive (disjoint) if they have no outcomes in common.
If E and F are mutually exclusive, then "E or F" includes all outcomes in E or F.
Notation: sample space with events E and F such that E ∩ F = Ø.
Addition Rule (for mutually exclusive events)
If E and F are mutually exclusive, then the empirical probability satisfies
P{E\,\text{or}\,F} = \frac{#\text{outcomes in } E\text{ or } F}{\text{Total observations}} = \frac{#E + #F}{\text{Total observations}} = PE + P_F
Therefore, P{E\,\text{or}\,F} = PE + P_F \quad \text{if } E \text{ and } F \text{ are mutually exclusive}
Non-occurrence Rule (Complement Rule)
If A is an event, the probability that A does not occur is the complement of A:
P(A^c) = 1 - P(A)
This rule is often called the complement formula.
Back to the Warm-up (recap of the counting approach)
Setup
Population belongs to exactly one of four groups: A, B, C, D.
Sub-population composition: 1 person from A, 5 from B, 2 from C, 4 from D.
Total outcomes: 1 + 5 + 2 + 4 = 12
Uniform Probability Formula (counting form)
PA = \frac{1}{12}, \quad PD = \frac{4}{12} = \frac{1}{3}
Probabilities for combined events
P_{A\,\text{or}\,D} = \frac{1 + 4}{12} = \frac{5}{12}
P_{B\,\text{or}\,C} = \frac{5 + 2}{12} = \frac{7}{12}
Addition Rule check
A and D are mutually exclusive (a person belongs to exactly one group).
Therefore, by the Addition Rule for mutually exclusive events,
P{A\,\text{or}\,D} = PA + P_D = \frac{1}{12} + \frac{4}{12} = \frac{5}{12}
Quick references (formulas to remember)
Empirical Probability Formula
P_E = \frac{#\text{favorable outcomes for } E}{\text{Total # of outcomes in the sample space}}
Uniform Probability Formula
(Same form as empirical; used when all outcomes are equally likely.)
P_E = \frac{#\text{favorable outcomes for } E}{\text{Total # of outcomes in sample space}}
Addition Rule (Mutually Exclusive)
P{E\,\text{or}\,F} = PE + P_F \,\text{ if } E \cap F = \emptyset
In terms of counts:
P_{E\,\text{or}\,F} = \frac{#E + #F}{\text{Total observations}}
Non-occurrence / Complement Rule
P(A^c) = 1 - P(A)
Example recap: Goldfinch probability
P_{goldfinch} = \frac{8}{15 + 17 + 8 + 7} = \frac{8}{47} \approx 0.17
Example recap: Empirical vs theoretical interpretation
Empirical: relative frequency based on data (47 observations in Maria’s birdwatching).
Theoretical: uses a model; the same data can inform or validate a model.
Important notes
Empirical probabilities are estimates; theoretical probabilities are based on assumptions/models.
Both approaches can yield the same numerical result in simple cases, but their interpretations differ.
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