S1 3.1, S1 3.2 - Events and sample spaces

S1 3.1, S1 3.2 - Events and sample spaces

Use elementary probabilities in S1 3.1 and sample spaces, complements, exclusive events, unions and intersections in S1 3.2. Translate “not”, “and” and “or” into precise events, then count outcomes or combine supplied probabilities without counting an overlap twice.

Elementary probabilities

When you roll a fair die, “a multiple of three” and “an even number” can both happen at once. How can you find the probability of either event without counting a six twice? We will build the event language before combining it.

An outcome is a possible result of the experiment at the level of detail being recorded. An event is a collection of outcomes meeting a description. For a die, the event “more than four” is {5,6}\{5,6\}.

Probability

The probability P(A)P(A) measures the chance of event AA, on a scale from 00 to 11. An impossible event in the sample space has probability 00; a certain event has probability 11. The probabilities of all separate possible outcomes sum to 11.

If the outcomes are equally likely, count them:

P(A)=number of outcomes in Atotal number of outcomes.P(A)=\frac{\text{number of outcomes in }A}{\text{total number of outcomes}}.

This counting rule is not printed in the formula booklet: learn it. The equal-likelihood condition matters. If outcomes have different probabilities, add the probabilities of the separate outcomes belonging to the event instead.

Worked example 1: equally likely outcomes

Question. A fair six-sided die is rolled. Find the probability of a result greater than 44.

Plan. List the favourable outcomes and divide by the number of equally likely outcomes.

A={5,6},P(A)=26=13.A=\{5,6\},\qquad P(A)=\frac26=\frac13.

Answer. 1/31/3.

Check. Each of 55 and 66 has probability 1/61/6, so adding them also gives 1/31/3.

Change only the probabilities of the six outcomes: the die is now biased, while the event is unchanged.

Worked example 2: unequal probabilities

Question. A biased die has probabilities 0.10,0.10,0.20,0.20,0.15,0.250.10,0.10,0.20,0.20,0.15,0.25 for results 1,2,3,4,5,61,2,3,4,5,6, respectively. Find the probability of a result greater than 44.

Plan. Add the supplied probabilities of 55 and 66, rather than counting labels.

P(A)=0.15+0.25=0.40.P(A)=0.15+0.25=0.40.

Answer. 0.40.4, not 2/62/6.

Check. The probabilities of all six results sum to 11. The other four sum to 0.600.60, leaving 0.400.40 for this event.

A probability describes the model's chance, whereas an observed relative frequency describes the actual proportion in a set of trials. Relevant repeated observations can help estimate an unknown probability, but the two need not match exactly in a finite sample.

Watch out

“Two favourable labels out of six” gives 2/62/6 only when the six labels are equally likely. A biased device can have the same possible outcomes and different probabilities.

Use the supplied probabilities for a fresh event.

Sample spaces and complements

A sample space makes it clear which results are possible and which lie outside a chosen event.

Sample space and complement

The sample space SS is the set of all possible outcomes. The complement A′A' contains all outcomes in SS that are not in AA. An outcome lies in exactly one of AA and A′A'.

For example, record a coin result H or T and a spinner result 1,2,31,2,3. The sample space is the six ordered pairs below. The order tells you which result came from which device.

Spinner 11Spinner 22Spinner 33
Coin H(H,1)(\mathrm{H},1)(H,2)(\mathrm{H},2)(H,3)(\mathrm{H},3)
Coin T(T,1)(\mathrm{T},1)(T,2)(\mathrm{T},2)(T,3)(\mathrm{T},3)

The list itself does not prove that its entries are equally likely. State or use the chance model supplied. A sample-space table is useful for a small experiment because it prevents missing or repeating an outcome.

Complement rule

P(A′)=1−P(A).P(A')=1-P(A).

This follows because AA and A′A' together fill the sample space without overlap. This rule is not printed in the formula booklet: learn it.

A complement must refer to the complete event description. The complement of “at least one head” is “no heads”, rather than “at least one tail”. A result can contain both a head and a tail.

Worked example 3: listing a complement

Question. The six coin-spinner pairs in the table are equally likely. Let AA be “the spinner shows 33”. List A′A' and find its probability.

Plan. Remove the two outcomes belonging to AA, keeping every other outcome.

A′={(H,1),(H,2),(T,1),(T,2)},A'=\{(\mathrm H,1),(\mathrm H,2),(\mathrm T,1),(\mathrm T,2)\}, P(A′)=46=23.P(A')=\frac46=\frac23.

Answer. The complement is “the spinner shows 11 or 22”, with probability 2/32/3.

Check. P(A)=2/6=1/3P(A)=2/6=1/3, so 1−P(A)=2/31-P(A)=2/3.

Change only the event, from “shows 33” to “shows at least 22”. Its complement must change with the full description.

Worked example 4: complement of an inequality

Question. In the same experiment, let BB be “the spinner shows at least 22”. List B′B' and find P(B′)P(B').

Plan. Translate “not at least 22” into “less than 22”, then read the sample space.

B′={(H,1),(T,1)},P(B′)=26=13.B'=\{(\mathrm H,1),(\mathrm T,1)\},\qquad P(B')=\frac26=\frac13.

Answer. “The spinner shows 11”, with probability 1/31/3.

Check. Four of the six pairs have spinner result 22 or 33, so P(B)=2/3P(B)=2/3 and the two probabilities add to 11.

An exhaustive list must include outcomes outside the event as well as those inside it.

Mutually exclusive events

Two events are mutually exclusive when a single outcome cannot belong to both.

Mutually exclusive events

Events AA and BB are mutually exclusive if they have no outcomes in common. Their intersection is empty, so P(A∩B)=0P(A\cap B)=0. For such events,

P(A or B)=P(A)+P(B).P(A\text{ or }B)=P(A)+P(B).

This is the no-overlap case of the addition rule in the formula booklet.

On one die roll, “odd” and “even” are mutually exclusive and together include all outcomes. “Odd” and “six” are also mutually exclusive, but together they omit 22 and 44. Thus mutually exclusive does not by itself mean complementary. Complementary events are both non-overlapping and exhaustive.

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