3. Basic probability concepts

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A consequence of the frequency definition of probability is that, for a frequentist, probability is defined as a state of nature that is independent of a person’s cognitions or beliefs. As a result, if two frequentists disagree about the value of a probability, at least one of them must be wrong. Subjective probabilities Unlike classical and empirical probabilities, which are derived either from knowledge of potential outcomes or from data, a subjective (or degree-of-belief or subjectivist) probability describes an individual’s personal judgment about how likely a particular event is to occur. It is based not on a precise computation, but rather on a reasonable assessment of the situation by a knowledgeable person. Subjective probabilities can be based on qualitative factors, previous experience in similar situations, and intuition. The subjectivist interpretation of probability differs from the frequentist interpretation in two important ways. First, the subjectivist defines probability as a belief an observer has about nature, rather than as a state of nature independent of an observer. Second, subjectivist’s beliefs about the value of a probability may differ for different observers. For example, meteorologists often assign probabilities to particular weather conditions, such as “The probability of rain tomorrow is 20%”. Although meteorological models are becoming increasingly sophisticated and may be used to estimate frequentist probabilities by running thousands of trials, traditionally a meteorologist would consider winds, pressure areas and fronts, adjacent weather systems, and perhaps a decent amount of intuition to estimate the probability of rain. The estimate would apply implicitly to a specific area of interest (domain). The probability of rain on the surface of the earth is likely to be close to 1, and in my back yard it may be close to 0, while still being 20% for the region in which I live. Regardless of how the probability is estimated, we typically use the information subjectively to make decisions about how to dress and whether to carry rain gear. If I’m walking, I might decide to carry a coat and umbrella if the probability of rain is greater than 50%. If I’m walking and wearing a suit, I might decide to carry a coat and umbrella if the probability of rain is greater than 60%. If I’m driving, however, I might skip the coat and grab an umbrella only if the probability is greater than 70% or 80%. My decision would be based on past experience. A person more concerned about getting wet might carry an umbrella whenever the probability of rain is greater than 30%. As an example of how the assessment of a subjective probability might differ for different individuals, consider the case of three people who have different information about the outcome of a specific toss of a fair coin. Mary, who saw the flip of the coin come up heads, would say the probability that the coin landed heads is 1. Javier, who didn’t see the outcome of the coin flip, might say that the probability that the coin landed heads is ½. And Xiao, who didn’t see the coin flip but did see the look on Mary’s face immediately following the flip, might say that the probability that the coin landed heads is ¾. Under the subjectivist definition of probability all three individuals would be equally correct as long as their beliefs were consistent with the axioms of probability. Although their quality depends on the experience of the person producing them, subjective probabilities can be perfectly acceptable estimates of “true” probabilities. However, whether 8
subjective probabilities should be used in further statistical analyses is a subject of some controversy. Statisticians of the frequentist (also called objectivist or classical) school of thought reject their use, while those of the bayesian school accept them. We will discuss this contrast in more detail below. Compound events Multiplication rule for simultaneous probabilities Addition rule Statistical independence Counting rules: permutations and combinations Conditional probabilities Odds ratios Likelihood and maximum likelihood Bayesian probabilities Bayes’ theorem Random variables 9
Page 1 and 2: From: R.E. Strauss, Statistical Hyp
Page 3 and 4: (observations or entities) from an
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3. Basic probability concepts

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