Useful properties of probability

We now look at some of the consequences of the three axioms of probability given in the previous section.

An event space ε depicted as a region enclosed in rectangle and the event A and B where A is contained in B.
Event \(A\) is a subset of event \(B\).
Detailed description

Property 1

If the event \(A\) is a subset of the event \(B\), then \(\Pr(A) \leq \Pr(B)\). That is,

\[ A \subseteq B \ \implies \ \Pr(A) \leq \Pr(B). \]

This property has a useful application. We write \(A \Rightarrow B\) to mean that, if \(A\) occurs, then \(B\) occurs. So \(A \Rightarrow B\) is the same as \(A \subseteq B\), which implies \(\Pr(A) \leq \Pr(B)\). For example:

so \(A \subseteq B\) and therefore \(\Pr(A) \leq \Pr(B)\); his chance of finding the treasure is at most equal to his chance of solving the puzzle.

While property 1 is important, it is rather obvious.

Property 2
\(0 \leq \Pr(A) \leq 1\), for each event \(A\).

This property formalises the scale for probabilities, given the axioms.

An event space ε depicted as a region enclosed in rectangle and the event C not containing any crosses and labelled as the empty set.
The empty event.
Detailed description

Property 3
\(\Pr(\varnothing) = 0\).

It would be rather strange if the probability of the empty set was anything other than zero, so it is reassuring to confirm that this is not so: \(\Pr(\varnothing) = 0\), as expected.

An event space ε depicted as a region enclosed in rectangle and the event A. The region in ε but not in A labelled as A complement.
An event \(A\) and its complementary event \(A'\).
Detailed description

Property 4
\(\Pr(A') = 1 - \Pr(A)\), for each event \(A\).

This property is surprisingly useful and is applied frequently. It is most effective when the probability of the event of interest is difficult to calculate directly, but the probability of the complementary event is known or easily calculated.

Exercise 3

Suppose that, in a four-child family, the probability of all four children being boys is 0.07. What is the probability that a four-child family contains at least one girl?

Property 5 (Addition theorem)

\(\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)\), for all events \(A\) and \(B\).

An event space ε depicted as a region enclosed in an elliptical type loop and three events A, B and C depicted as regions within the region of  ε. Each of the regions labelled.
Detailed description
\(B\) \(B'\)
\(A\) \(A \cap B\) \(A \cap B'\)
\(A'\) \(A' \cap B\) \(A' \cap B'\)
Two representations of the addition theorem.

Both the diagram and the table above give representations of the event space \(\mathcal{E}\) in terms of the two events \(A\) and \(B\) and their complements.

The impression we get from the diagram, from the table, or just using basic logic is that, if we add together the probability of \(A\) and the probability of \(B\), then \(A \cap B\) is included in both events, so it has been counted twice. Hence, to obtain the probability of at least one of \(A\) and \(B\) occurring, that is, to obtain \(\Pr(A \cup B)\), we need to subtract \(\Pr(A \cap B)\) from \(\Pr(A) + \Pr(B)\). This justifies the formula given by the addition theorem (property 5). We can demonstrate the result more formally, as follows.


Exercise 4

Lego sells `minifigures'. There are 16 distinct minifigures, and they are distributed at random among shops (it is said). Furthermore, the minifigures cannot be identified prior to purchase and removal of the packaging. A young child wishes to obtain the `Pirate Captain', one of the 16 figures in Series 8. She persuades her parent to buy a minifigure at two different shops. Define the events:

Assume that the distribution of the minifigures is random across shops, and assume that \(\Pr(A \cap B) = \dfrac{1}{256}\).

  1. What is \(\Pr(A)\)? \(\Pr(B)\)?
  2. What is the probability of the child's wishes being satisfied?
  3. Describe each of the following events in words:
    1. \(B'\)
    2. \(A' \cup B'\)
    3. \(A \cap B'\)
    4. \(A' \cap B'\).
  4. Which event is more probable: \(A \cup B\) or \(A' \cap B'\)?

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