Sesión 10

We begin this section by presenting three simple, self-evident truths known asaxioms which list the basic properties we require of event probabilities.

4.9.1 Axioms of probability

1. For any event A, 0 ≤ P (A) ≤ 1.

2. For sample space S, P (S)= 1.

3. If {Ai}, i= 1, . . . , n, are mutually exclusive events, then the probability of their ‘union’ is the sum of their respective probabilities, i.e.

The first two axioms should not be surprising. The third may appear a little more daunting. Events are labelled mutually exclusive when they cannot both simultaneously occur. For example, when rolling a die once the event A = ‘get an even score’ and B = ‘get an odd score’ are mutually exclusive.

Extending this, a collection of events is pairwise mutually exclusive if no two events can happen simultaneously. For instance the three

events A, B and C are pairwise mutually exclusive if A and B cannot happen togetherand B and C cannot happen together and Aand C cannot happen together. Another way of putting this is that a collection of events is pairwise mutually exclusive ifat most one of them can happen.

Related to this is the concept of a collection of events being collectively exhaustive. This means at least one of them must happen, i.e. all possible experimental outcomes are included amongst the collection of events.

4.9.2 Notational vocabulary

Axiom 3 above introduced a new symbol. For the remainder of this chapter, various symbols connecting sets will be used as a form of notational shorthand. It is important to be familiar with these symbols, hence two ‘translations’ are provided — one for kids and

one for grown-ups.^{8 8}I’ll leave you to decide which one of

these mutually exclusive and exhaustive sets applies to you.

Symbol ‘Kids’ version ‘Adult’ version Example

∪ or union A ∪ B= ‘A union B’

∩ and intersect A ∩ B= ‘A intersect B’

c not complement of A^c= ‘complement of A’

| given conditional on A|B= ‘A conditional on B’

Also do make sure that you distinguish between a set and the probability of a set. This distinction is important. A set, remember, is a collection of elementary outcomes from S, whereas a probability (from axiom 1) is a number from the unit interval, [0, 1]. For example A = ‘an even die score’, while P (A)= 0.5, for a fair die.

4.9.3 Venn diagrams

The previous coin and die examples have been rather simple (for illustrative purposes). Hence it is highly likely (in fact with a probability of 1!) that you will encounter more challenging sets and sample spaces. Fear not — there is a helpful geometric technique which can often be used: we represent the sample space elements in aVenn diagram.

Imagine we roll a die twice and record the total score. Hence our sample space will be

S= {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Suppose we are interested in the following three events:

A= an even total, that is A = {2, 4, 6, 8, 10, 12};

B= a total strictly less than 8, that is B = {2, 3, 4, 5, 6, 7};

C= a total greater than 4 but less than 10, that is C= {5, 6, 7, 8, 9}.

Having defined these events, it is therefore possible to insert every element in the sample space S into a Venn diagram:

A B

10, 12

3

11

C

6

9 2, 4

8

5, 7

Figure 4.1: Venn diagram for pre-defined sets A, B and C recording the total score when a die is rolled twice.

The box represents S, so every possible outcome of the experiment (total score when a die is rolled twice) appears within the box.

Three (overlapping) circles are drawn representing the events A, B and C. Each element of S is then inserted into the appropriate area.

For example, the area where the three circles all intersect represents the event A ∩ B ∩ C into which we place the element ‘6’, since this is the only member of S which satisfies all three events A, B and C.

Hence we can now determine the following sets, for example:

A ∩ B= {2, 4, 6} A ∩ C= {6, 8}

A ∩ B ∩ C= {6} (A ∪ B ∪ C)^c= {11}

A ∩ B ∩ C^c= {2, 4} A^c∩ B= {3, 5, 7}

(A ∪ C)^c∩ B= {3} A|C= {6, 8}.

4.9.4 The additive law

Let A and B be any two events. The additive law states that P(A ∪ B)= P (A) + P (B) − P (A ∩ B).

So P (A ∪ B) is the probability that at least one of A and B occurs, and P (A ∩ B) is the probability that both A and B occur.

Let’s think about this using a Venn diagram. The total area of the Venn diagram overleaf is assumed to be 1, so area represents probability. Event A is composed of all points in the left-hand circle, and event B is composed of all points in the right-hand circle. Hence

P(A) = area x + area z P(B) = area y + area z

P(A ∩ B) = area z P(A ∪ B) = area x + area y + area z.

A

x z y

B

Figure 4.2: Venn diagram illustrating the additive law.

P(A ∪ B) = P (A) + P (B) − P (A ∩ B)

= (area x + area z) + (area y + area z) − (area z)

= area x + area y + area z.

Hence, to compute P (A ∪ B) we need to subtract P (A ∩ B) otherwise that region would have been double-counted.

Activity

Consider an industrial situation in which a machine component can be defective in two ways such that

P(defective in first way)= 0.01 P(defective in second way)= 0.05 P(defective in both ways)= 0.001

Then it follows that the probability that the component is defective is

0.01+ 0.05 − 0.001 = 0.059.

Special cases

Two important special cases of the additive law are worth noting:

1. If A and B are mutually exclusive events, i.e. they cannot occur simultaneously, then P (A ∩ B)= 0. Hence

P(A ∪ B) = P (A) + P (B) − P (A ∩ B)

= P (A) + P (B) − 0

= P (A) + P (B).

Such events can be depicted by two non-overlapping sets in a Venn diagram as in Figure 4.3.

A B

Figure 4.3: Venn diagram illustrating two mutually exclusive events.

Now re-visit axiom 3, to see this result generalised for n mutually exclusive events.

2. The probability of an event Anot happening, i.e. the complement, A^c, is

P(A^c)= 1 − P (A).

4.9.5 The multiplicative law

This rule is concerned with the probability of two events happening at the same time — specifically when the two events have the special property ofindependence.

An informal definition of independence is that two events are said to be independent if one has no influence on the other. Formally, events A and B are independent if the probability of their intersect is the product of their individual probabilities, i.e.

P(A ∩ B)= P (A) · P (B).

For example consider rolling two fair dice. The score on one die has no influence on the score on the other. Hence the respective scores are independent events. Hence P (two sixes)= ¹₆×¹₆ = ₃₆¹.

Note the multiplicative (or product) law does not hold for

dependent events, which is the subject ofconditional probability, discussed shortly. Also, take a moment to ensure you are

comfortable with the terms ‘exclusive’ and ‘independent’. These are not the same thing, so do not get these terms confused.