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Free lesson · Statistics

Start with counts, probabilities and averages

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Start with the idea

  • Count possible outcomes first.
  • A probability measures how likely an event is.
Symbols, units & horizon
  • P(even): probability of an even outcome on the fair die
  • x̄: mean of the three values 2,4,6
  • /: division
  • die outcomes and probabilities: dimensionless in this example

When and why to use this

Use probabilities for uncertain events and averages to summarize observations. Weighted expectation, variance and sampling uncertainty build on these operations.

  • Use an explicitly fair six-sided die.
  • The even outcomes are 2, 4 and 6.
  • Three of six equally likely outcomes are even.
  • Probabilities lie between zero and one.
  • For mutually exclusive, exhaustive outcomes, probabilities sum to one.
  • An ordinary average adds equally weighted observations and divides by their count.
P(even)=36=12,x‾=2+4+63=4
Core rule · definition and worked arithmetic

Start with counts, probabilities and averages

  1. List the favorable outcomes: 2, 4, 6. There are three.
  2. Divide by all six equally likely outcomes: 3/6=1/2.
  3. For the separate average of the three even values, add 2+4+6=12 and divide by 3 to get 4.
Work it by hand

The probability .5 is not the average 4. They answer different questions: how likely versus how large.

Use the rule

  • Name the inputs and units.
  • Work the small example by hand.
  • Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Research sources, review dates and limitations

Capstone checkpoint 2 / Interpret uncertain payoffs

Synthetic exercise · self-assessed. Use the existing capstone expectancy walkthrough. Before calculating, predict what happens when round-trip costs increase. Reproduce the 22-basis-point expected return per exposed dollar.

Save in your practice notes: A hand calculation with probability, payoff and cost units. Open practice studio →

Check your reasoning

An expected return is not a guaranteed outcome or an account return; exposure and estimation uncertainty still matter.

Full hand-working, definitions and runnable Python →

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def fair_event_probability(favorable, total):
    if total <= 0 or not 0 <= favorable <= total: raise ValueError("Valid counts required")
    return favorable/total

def arithmetic_mean(values):
    if not values: raise ValueError("At least one value required")
    return sum(values)/len(values)

print(fair_event_probability(3,6), arithmetic_mean([2,4,6]))

Continue learning

Statistics & Probability — all lessons
  1. Start with counts, probabilities and averages
  2. Random outcomes, sample averages and the limits of the bell curve
  3. Conditional probability: update a belief with evidence
  4. Prediction markets: probability, price and net expected value
  5. Expected value: measure the payoff before choosing the risk
  6. Correlation: how many strategies do you really have?
  7. Conditional probability and Bayes: where the edge actually lives
  8. The central limit theorem: sampling means under explicit assumptions
  9. Estimation uncertainty and Bayesian updating

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations