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Remarks on the concept of "Probability"

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  • Mathematics

Abstract

Connexions module: m10953 1 Remarks on the concept of "Probability" ∗ David Lane This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Inferential statistics is built on the foundation of probability theory, and has been remarkably successful in guiding opinion about the conclusions to be drawn from data. Yet (paradoxically) the very idea of probability has been plagued by controversy from the beginning of the subject to the present day. In this section we provide a glimpse of the debate about the interpretation of the probability concept. One conception of probability is drawn from the idea of symmetrical outcomes. For example, the two possible outcomes of tossing a fair coin seem not to be distinguishable in any way that affects which side will land up or down. Therefore the probability of heads is taken to be 1 2 , as is the probability of tails. In general, if there are N symmetrical outcomes, the probability of any given one of them occurring is taken to be 1 N . Thus, if a six-sided die is rolled, the probability of any one of the six sides coming up is 1 6 . Probabilities can also be thought of in terms of relative frequencies. If we tossed a coin millions of times, we would expect the proportion of tosses that came up heads to be pretty close to 1 2 . As the number of tosses increases, the proportion of heads approaches 1 2 . Therefore, we can say that the probability of a head is 1 2 . If it has rained in Seattle on 62% of the last 100, 000 days, then the probability of it raining tomorrow might be taken to be 0.62. This is a natural idea but nonetheless unreasonable if we have further information relevant to whether will rain tomorrow. For example, if tomorrow is August 1, a day of the year on which it seldom rains in Seattle, we should only consider the percentage of the time it rained on August 1. But even this is not enough since the probability of rain on the next Augu

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