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description starts with the idea of a standard for uncertainty. He provides an analogy using the concept of balls in an urn. Initially, the balls are of two different colours, black and white. In all other respects, size, weight, texture, etc., they are identical. In particular, if one were to pick a ball from the urn, without looking at its colour, it would not be possible to tell what colour it was. The two colours of balls are in the urn in proportions
and
for black and white balls, respectively, such that
. For example, if there were 10 balls in the urn of which 6 were black and 4 were white, then
, and
.
The urn is shaken up and the balls thoroughly mixed. A ball is then drawn from the urn. Because of the shaking and mixing, it is assumed that each ball, regardless of colour, is equally likely to be selected. Such a selection process, in which each ball is equally likely to be selected, is known as a random selection, and the chosen ball is said to have been chosen at random.
The ball, chosen at random, can be either black, an event that will be denoted , or white, an event that will be denoted . There are no other possibilities; one and only one of these two events has to occur. The uncertainty of the event , the drawing of a black ball, is related to the proportion of black balls in the urn. If is small (close to zero), is unlikely. If is large (close to 1), is likely. A proportion close to 1/2 implies that and are about equally likely. The proportion is referred to as the probability of obtaining a black ball on a single random drawing from the urn. In a similar way, the proportion is referred to as the probability of obtaining a white ball on a single random drawing from the urn.
Notice that on this simple model probability is represented by a proportion. As such it can vary between 0 and 1. A value of occurs if there are no black balls in the urn, and it is, therefore, impossible to draw a black ball from the urn. The probability of obtaining a black ball on a single random drawing from the urn is zero. A value of occurs if all the balls in the urn are black. It is certain that a ball drawn at random from the urn will be black. The probability of obtaining a black ball on a single random drawing from the urn is one. All values between these extremes of 0 and 1 are possible (by considering very large urns containing very large numbers of balls).
A ball has been drawn at random from the urn. What is the probability that the selected ball is black? The event is the selection of a black ball. Each ball has an equal chance of being selected. The colours black and white of the balls are in the proportions and , respectively. The proportion, , of black balls corresponds to the probability that a ball, drawn in the manner described (i.e. at random) from the urn is black. It is then said that the probability a black ball is drawn from the urn, when selection is made at random, is . Some notation is needed to denote the probability of an event. The probability of , the drawing of a black ball, is denoted and similarly denotes the probability of the drawing of a white ball. Then it can be written that and . Note that
This concept of balls in an urn can be used as a reference for considering uncertain events. The methodology has been described as follows (Lindley 2006):
Your6 probability of the uncertain event of rain tomorrow is the fraction of [black] balls in an urn from which the withdrawal of a [black] ball at random is an event of the same uncertainty for you as that of the event of rain. [] You are invited to compare that event with the standard, adjusting the number of [black] balls in the urn until you have the same beliefs in the event and in the standard. Your probability for the event is then the resulting fraction of [black] balls. (p. 35)
Another example concerns a hypothetical sporting event. Let denote the uncertain event that the England football team will win the next major international football championship. Let denote the uncertain event that a black ball will be drawn from the urn. A choice has to be made between and , and this choice has to be ethically neutral. If is chosen and a black ball is drawn from the urn then a prize is won. If is chosen and England do win the championship the same prize is won. The proportion of black balls in the urn is known in advance. Obviously, if then is the better choice, assuming, of course, that England do have some non‐zero probability of winning the championship. If then is the better choice. Somewhere in the
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