Probability

1 Introduction

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In this chapter we will explain what we mean by probabilities, and look at a range of different calculations of probabilities that could be required of you in the exam.

Risk and uncertainty

Most business decisions are about the future, and the future is rarely known for certain. A new product may sell well or badly; next month’s demand may be high or low. Management accountants describe this in two ways.

Where risk can be measured, probabilities can be put on the outcomes and used to calculate an expected value (section 4). The more uncertain a situation is, the less reliable those probabilities – and the decisions based on them – become (section 5). In BA2 you will be asked to work with probabilities and expected values; you will not be asked to apply special techniques for decisions under uncertainty.

2 Simple probabilities

The probability of an event occurring is the likelihood or the chance that it will occur.

So, for example, if we toss a coin then there are two possible outcomes - that it falls as a head or as a tail - and both outcomes have an equal chance of occurring. Therefore the probability of it being a head - one of the two outcomes - is 1 in 2, which can be expressed as a probability of 1/2, or as 0.5, or as 50%.

Where every outcome is equally likely:

Probability of an event = number of outcomes that give the event ÷ total number of possible outcomes

A probability is always between 0 and 1. A probability of 0 means the event cannot happen; 1 means it is certain. If a list covers every possible outcome, their probabilities add up to 1 – one of them is certain to happen.

The addition rule – ‘either … or …’

When we want the probability that either one event or another happens, we add the probabilities.

• Mutually exclusive events cannot happen at the same time (a single die cannot show a 1 and a 2). Then P(A or B) = P(A) + P(B).

• Events that can happen together (a card can be an ace and a diamond). Adding the two probabilities would count the overlap twice, so it is subtracted once: P(A or B) = P(A) + P(B) − P(A and B).

A standard pack of 52 playing cards has four suits – hearts, diamonds, spades and clubs – each of 13 cards: ace, 2 to 10, jack, queen and king.

Probabilities, proportions and percentages

In business, probabilities usually come from past results. If an outcome has happened in a certain proportion of past cases, that proportion is used as its probability. A probability of 0.25 is the same as a proportion of 1/4 and a percentage of 25% – it means the outcome is expected to happen in a quarter of cases.

Part (c) is a conditional probability – the probability of one event (female) given that another has already happened (voting for Party A). The total we divide by is no longer the whole group of 200, only the 85 for whom the condition is true. Drawing up a table of the whole group, as in the answer, is the safest way to get it right.

3 Joint probabilities

If we want to know the probability of two or more events occurring, then we can calculate it by multiplying the individual probabilities together.

For example, if we toss a die two times, then there are 36 possible outcomes - a 1 and a 1, a 1 and a 2, a 1 and a 3, and so on. So the probability of getting two 6s is 1/36 because two 6s is just one of the 36 possible outcomes. However, we could have got the same answer by saying that the probability of getting a 6 on the first toss is 1/6; the probability of getting a 6 on the second toss is 1/6; and therefore the probability of getting two 6s is 1/6 x 1/6 = 1/36.

This is the multiplication rule: for independent events – where the result of one has no effect on the other, like two tosses of a die – P(A and B) = P(A) × P(B). Use it whenever the question wants one thing and another to happen.

In Example 5, we replaced the playing card after the first pick, and therefore there were 52 cards to pick from on both picks. However, if the first card had not been replaced, then it would change the probabilities for the second card.

Because the first card was not replaced, the probability for the second card depends on what happened first – the events are dependent, and the second probability is a conditional probability. The multiplication rule still applies, using the conditional probability for the second pick.

Do not confuse mutually exclusive with independent. Mutually exclusive events cannot happen together, so P(A and B) = 0. Independent events can happen together, and P(A and B) = P(A) × P(B).

4 Expected values

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Probabilities can be useful in a business situation where there are various possible results that could occur for which we know the probabilities. We can use the probabilities to arrive at an ‘average’ result, which we call the expected value.

Expected value (EV) = the sum of (each possible outcome × its probability)

It is a weighted average: each outcome is weighted by how likely it is. The probabilities used must add up to 1.

Expected values can be used to help make decisions, in which case it is common to need to produce a table first showing the possible outcomes and their probabilities.

5 The limitations of expected values

There are two main limitations as to the use of expected values in decision making:

• it is unlikely that the probabilities can be determined with accuracy, and should the probabilities turn out to be different then the wrong decision may have been made.

• for a ‘one-off’ decision, the actual outcome is unlikely to coincide with the expected outcome - it may turn out to be better or it may turn out to be worse.

In Example 7 the expected revenue is $310,000 a year, but no single year will actually produce $310,000. If the product turns out to be popular, demand may be high every year and revenue $500,000 every year; if it is unpopular, revenue may be $200,000 every year. The expected value is only a reliable guide when the decision is repeated many times, so that the good and bad outcomes average out. In the same way, John in Example 8 will never earn exactly $4,500 in a month – it is not one of the possible profits.

6 Test your knowledge

Two quick checks before you move on: work through the flashcards to fix this chapter’s terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.