100 topics · Read, check sources and follow your curiosity.

A little room to wonder

Probability & Chance

Build a complete set of outcomes, count an event and separate theoretical probability from a short run of results.

Are the possibilities you counted actually equally likely?

Four coloured six-sided dice with different faces visible.
A fair-dice probability model assumes equally likely faces. A photograph of dice cannot establish that fairness.Diacritica, via Wikimedia Commons. CC BY-SA 3.0. · Source · CC BY-SA 3.0

State the experiment and its outcomes

A sample space is the set of possible outcomes for a defined chance experiment. An event is a selection of those outcomes. When all elementary outcomes are equally likely, an event’s probability is its number of favourable outcomes divided by the total. The equal-likelihood condition is essential, not a decorative phrase.

Probabilities run from 0 to 1 in the model. A value of 1/2 does not mean outcomes must alternate. It describes the chance assigned to an event under the assumptions, rather than a promise about the next few observations.

Work through two fair independent coins

For two labelled fair coins tossed independently, the ordered outcomes are HH, HT, TH and TT. Each has probability 1/4. Exactly one head corresponds to HT and TH, so its probability is 2/4 = 1/2. At least one head includes HH as well, giving 3/4.

The totals zero, one and two heads are not equally likely merely because there are three totals. The middle total contains two elementary outcomes. Listing the underlying outcomes prevents that counting mistake.

Replacement changes a model

Imagine a fictional bag with four red and two blue counters mixed so each counter is equally likely to be selected. The first selection has probability 2/6 = 1/3 of blue. If a blue counter is removed and not returned, the next selection is from four red and one blue, making its conditional probability of blue 1/5.

If the first counter is returned and the bag remixed under the same assumptions, the composition returns to four red and two blue. State which procedure you mean before calculating. A changed procedure can invalidate an otherwise correct fraction.

A short run is not a verdict

A small set of observed outcomes may differ substantially from theoretical proportions. Four heads in a row does not make a fair independent coin owe a tail. If real observations raise a question about the model, investigate the setup and collect suitable evidence rather than inventing a balancing rule.

Only if you feel like trying

Count the underlying possibilities

  1. List HH, HT, TH and TT and mark exactly one head and at least one head.
  2. Explain why the three possible head totals do not each have probability 1/3.
  3. Compare the fictional bag’s second selection with and without replacement after a blue first selection.

Optional paper calculation. No coin tosses, gambling, purchases or collection of real participant results is required.