Events

Some experiments always give the same result: burn magnesium in air and you always get magnesium oxide. Others are random: you know every result that could happen, but not which one will. Tossing a coin, rolling a die and drawing a card are random experiments.

The set of all possible outcomes is the sample space S.

• One coin: {H, T}. Two coins: {HH, HT, TH, TT}. With n coins there are 2ⁿ outcomes; with n dice, 6ⁿ (36 for two dice).
• Three coins, listed so nothing is missed: HHH; then one T moved through each place: HHT, HTH, THH; then two T's: HTT, THT, TTH; then TTT. That is 8 = 2³.

Experiments in stages are listed branch by branch. Toss a coin; if it shows H, toss it again; if it shows T, roll a die:

• S = {HH, HT, T1, T2, T3, T4, T5, T6}, which has 8 outcomes.
• Follow each branch to its end before starting the next one.

An event is any subset of the sample space; it happens if the outcome lands in it. Rolling a die: A = 'more than 4' = {5, 6}, B = 'even' = {2, 4, 6}. A die has 2⁶ = 64 possible events, one for each subset.

• Impossible event: φ, such as 'more than 6'. Sure event: S itself, such as 'less than 7'.
• Simple event: one outcome, such as 'an even prime' = {2}. Compound event: more than one outcome, such as B.

Words become set operations:

Mutually exclusive events cannot happen together: A ∩ B = φ. Rolling a die, 'even' and 'odd' are mutually exclusive; 'more than 4' and 'even' are not, because 6 is in both.

Events are exhaustive if together they cover the whole sample space: their union is S. On a die, E₁ = {1, 2}, E₂ = {3, 4}, E₃ = {5, 6} are exhaustive, and since no two overlap they are mutually exclusive and exhaustive.

With two dice, 'the sum is 2' = {(1, 1)} and 'the sum is 12' = {(6, 6)} are mutually exclusive but not exhaustive: most outcomes are in neither.