probability - 1 Probability 1.1 Introduction Bayesian...

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1 Probability 1.1 Introduction Bayesian thinking is based on the subjective viewpoint of probability. In this chapter, we will talk about the different ways of thinking about probability. 1.2 Measuring Uncertainty We live in a world of uncertain events and some events are more likely to occur than other events. We use words such as “likely”, “probable”, “possible”, “rare”, and “maybe” to describe this uncertainty. It is natural to use numbers that we call probabilities to quantify this uncertainty. The probability of an event A , denoted P ( A ), is a number between 0 and 1 assigned to the event A where a larger number indicates that the event is more likely to occur. Some uncertain events already have numbers assigned to them. In games of chance where dice are rolled or cards are dealt from a well-shuffled deck, outcomes have particular probabilities. For example, the chance of rolling two dice equal to double-sixes is 1/36 and the chance of dealing a four of diamonds in a regular deck is 1/52. In actuarial tables, there are assigned probabilities that a person’s life span will be a particular length based on one’s gender and age. These actuarial tables are used by insurance companies to write up a life insurance policy and decide on the cost of the policy to the customer. There are two ways of viewing probabilities that allow us to assign proba- bilities in games of chance and actuarial tables. The classical or“equally-likely” view assumes that one can represent the outcomes of a random experiment in such a way such that the outcomes are equally likely. Then each outcome is assigned a probability equal to one divided by the total number of outcomes. In the dice example, there are 36 equally likely ways of representing the pos- sible rolls of the dice, and the probability of one outcome (two sixes) is equal 1/36. In the card example, there are 52 possible draws in a deck of cards,
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2 1 Probability and if the deck is well-shuffled, each outcome such as “four of diamonds” is assigned the probability of 1/52. A second way of thinking about probabilities is based on long-run relative frequencies. Suppose you are able to repeat a random experiment many times under similar conditions. Then the probability of an event is approximated by its relative frequency in the large number of trials. This viewpoint can be applied in games of chance. For example, the probability that the sum of two dice is equal to 7 can be approximated by the relative frequency of 7 in many rolls of the two dice. This definition can also be used for actuarial tables. The chance that a male of age 70 will survive ten years can be approximated by the relative frequency of 70-year old males who survive ten years. Is it possible use the relative frequency viewpoint to measure uncertainty
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This note was uploaded on 01/01/2011 for the course STAT 665 taught by Professor Albert during the Spring '10 term at Bowling Green.

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probability - 1 Probability 1.1 Introduction Bayesian...

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