#### INDEPENDENT AND CONDITIONAL EVENTS

An independent event occurs when the outcome does not depend on the outcome of a previous event. For example, assuming that a dice is unbiased, then the probability of throwing a five on the second throw does not depend on the outcome of the first throw.

In contrast, with a conditional event, the outcomes of two or more events are related, ie the outcome of the second event depends on the outcome of the first event. For example, in Table 1, the company is forecasting sales for the first year of the new product. If, subsequently, the company attempted to predict the sales revenue for the second year, then it is likely that the predictions made will depend on the outcome for year one. If the outcome for year one was sales of $1,500,000, then the predictions for year two are likely to be more optimistic than if the sales in year one were $500,000.

The availability of information regarding the probabilities of potential outcomes allows the calculation of both an expected value for the outcome, and a measure of the variability (or dispersion) of the potential outcomes around the expected value (most typically standard deviation). This provides us with a measure of risk, which can be used to assess the likely outcome.

#### EXPECTED VALUES AND DISPERSION

Using the information regarding the potential outcomes and their associated probabilities, the expected value of the outcome can be calculated simply by multiplying the value associated with each potential outcome by its probability. Referring back to Table 1, regarding the sales forecast, then the expected value of the sales for year one is given by:

Expected value

= ($500,000)(0.1) + ($700,000)(0.2)

+ ($1,000,000)(0.4) + ($1,250,000)(0.2)

+ ($1,500,000)(0.1)

= $50,000 + $140,000 + $400,000

+ $250,000 + $150,000

= $990,000

In this example, the expected value is very close to the most likely outcome, but this is not necessarily always the case. Moreover, it is likely that the expected value does not correspond to any of the individual potential outcomes. For example, the average score from throwing a dice is (1 + 2 + 3 + 4 + 5 + 6) / 6 or 3.5, and the average family (in the UK) supposedly has 2.4 children. A further point regarding the use of expected values is that the probabilities are based upon the event occurring repeatedly, whereas, in reality, most events only occur once.

In addition to the expected value, it is also informative to have an idea of the risk or dispersion of the potential actual outcomes around the expected value. The most common measure of dispersion is standard deviation (the square root of the variance), which can be illustrated by the example given in Table 2above, concerning the potential returns from two investments.

To estimate the standard deviation, we must first calculate the expected values of each investment:

Investment A

Expected value = (8%)(0.25) + (10%)(0.5) + (12%)(0.25) = 10%

Investment B

Expected value = (5%)(0.25) + (10%)(0.5) + (15%)(0.25) = 10%

The calculation of standard deviation proceeds by subtracting the expected value from each of the potential outcomes, then squaring the result and multiplying by the probability. The results are then totalled to yield the variance and, finally, the square root is taken to give the standard deviation, as shown in Table 3.