Modelling a random experiment

Sample space, events and probability distributions

An experiment is random when, repeated under the same conditions, it does not always give the same result. Before computing any probability, it must be modelled — that is, we must decide precisely what we are talking about.

The sample space

The sample space, written Ω, is the set of all possible results of the experiment. Each result is called an outcome.

Experiment                        Sample space Ω                Outcomes
-------------------------------   ---------------------------   ---------
tossing a coin                    { Heads ; Tails }                 2
rolling a six-sided die           { 1 ; 2 ; 3 ; 4 ; 5 ; 6 }         6
rolling two dice (ordered pair)   { (1;1) ; (1;2) ; ... ; (6;6) }  36
drawing a card from a 52-card deck  the 52 cards                   52

The choice of sample space is not innocent: for two dice, if only the sum is kept, the space becomes {2, 3, …, 12}, that is 11 outcomes — but they are not equally likely. Almost every error in this chapter comes from a badly chosen sample space.

Events

An event is a subset of the sample space: a set of outcomes that "realise" something.

A = "getting an even number"            A = { 2 ; 4 ; 6 }
B = "getting at least 5"                B = { 5 ; 6 }

The vocabulary follows that of sets:

Notation      Name                    Reads as
------------  ----------------------  ---------------------------------
A ∩ B         intersection            "A AND B" both occur
A ∪ B         union                   "A OR B" (at least one of the two)
Ā             complementary event     "A does NOT occur"
A ∩ B = ∅     mutually exclusive      they cannot happen together

        Ω
   ______________________
  |    ___       ___     |
  |   / A \_____/ B \    |        A ∩ B: the shared area
  |  |    (     )    |   |        A ∪ B: both discs together
  |   \___/-----\___/    |        Ā    : everything outside A
  |______________________|

Beware a linguistic trap: the mathematical "or" is inclusive. A ∪ B occurs when A alone, B alone, or both occur.

The probability distribution

Defining a probability distribution means assigning each outcome a number between 0 and 1, so that all these numbers add up to 1.

outcome    1     2     3     4     5     6
p(outcome) 1/6   1/6   1/6   1/6   1/6   1/6      total = 1   ✔

The probability of an event is then the sum of the probabilities of its outcomes.

A loaded die is modelled just as well: nothing requires the outcomes to be equally likely.

outcome    1     2     3     4     5     6
p(outcome) 0.1   0.1   0.1   0.1   0.1   0.5      total = 1   ✔  (loaded die)

The equally likely case

When all outcomes have the same probability — we speak of equiprobability — everything reduces to counting:

             number of outcomes favourable to A
  P(A)  =  --------------------------------------
                number of possible outcomes

This is the chapter's most-used formula, and the reason knowing how to count is indispensable.

Die: P("even") = 3/6 = 1/2
Two dice: P("sum 7") = 6/36 = 1/6
          (the 6 pairs (1;6), (2;5), (3;4), (4;3), (5;2), (6;1))

It applies only if equiprobability is justified: a "fair" die, a draw "at random". Without that assumption the formula is wrong.

Summary