# Essay Sample on Mathematical Probability

2021-07-08
3 pages
704 words
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Harvey Mudd College
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Course work
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Mathematical probability began with the picture of games of chance. Conditional probabilities are contingent on a previous result, or we can say the Conditional probability is a measure of the likelihood of an event given that (by assumption, presumption, assertion or evidence) another event has occurred. If the case of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B," or "the likelihood of An under the condition B," is usually written as P(A|B), or sometimes PB(A). For example, suppose you are drawing three balls - red, blue and green - from a bag. Each ball has an equal chance of being drawn. What is the conditional probability of drawing the red ball after already drawing the blue one? First, the likelihood of drawing a blue ball is about 33% because it is one possible outcome out of three. Assuming this first event occurs, there will be two balls remaining, with each having a 50% of being drawn. So, the chance of drawing a blue ball after already drawing a red ball would be about 16.5% (33% x 50%).

Are these three answers real ones, however? I presume we all agree that the first of them is correct. But the other two answers are very suspect. Indeed. The dual function of this essay is (firstly) to demonstrate that both are wrong. In that additional information is required to perform the updating reliably; then (secondly) to provide a different, and weaker, rule, that uses the appropriate extra information, to provide a convincing upgrade. Both these points have been suggested elsewhere, but with varying degrees of conviction. My task is to pursue the question systematically.

To catch a preliminary glimpse of why I take these views. Suppose that the existence of a blue ball is revealed by simply randomly picking a ball from the bag. Then in a long run of such trials, the time the ball was picked from the bag where a blue ball was picked will contain two balls. So it seems that a rational observer. Who finds out both that and how a blue ball has been selected, should now allocate the value ~ to the probability that the bag has two blue balls. On the other hand, suppose the new knowledge had been acquired quite differently. By someone revealing the existence of a red ball whenever possible. Such an informant would only indicate that the bag contained a blue ball if it were impossible to show that it contained in red. So after thus finding out that the chosen ball was blue. The probability that the next ball will be a red ball would surely jump to 1.

A thoroughly adequate description of the process of conditioning is, however, difficult to formulate. It is important to note (with e.g. Rosenkrantz . pp. 48-52) that as soon as we pose the sort of question the Rule of Conditionalization is routinely supposed to answer. We are envisaging two entirely separate types of activity as taking place. The first is typically a chance process in which a physical system evolves 'non-deterministically' from some initial microstate to one of a range of final states like the throw of a die which ends up in one of six possible orientations.

In contrast, I claim we need to attend to the probabilities associated with the outcomes of the second chance-process, I have called them, the chances of receiving the conditioning information. The need to embrace these possibilities is widely recognized in practice, for computations of posterior probabilities do in fact often use understandings of the second chance-process (as we will see in the next section). But the fact that it is these probabilities which are being used is often obscured, for the distinction between the two processes is blurred, especially in theoretical discussion. So the central problem confronted in this debate is one of articulation, finding the right description of procedures that are already in partial use.

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