{"id":406,"date":"2019-08-17T13:28:00","date_gmt":"2019-08-17T13:28:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=406"},"modified":"2026-01-15T19:07:32","modified_gmt":"2026-01-15T19:07:32","slug":"defining-properties-probability","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/defining-properties-probability\/","title":{"rendered":"Defining Properties of a Probability"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Probability Concepts (2021 Level I CFA\u00ae Exam \u2013 Reading 8)\",\n  \"description\": \"This video lesson covers the essential concepts of probability, including random variables, outcomes, events, and probability calculations. It explains different types of probability, such as empirical, subjective, and a priori, and covers key principles like unconditional vs. conditional probability, joint probability, and the use of tree diagrams and Bayes\u2019 theorem in decision-making.\",\n  \"uploadDate\": \"2020-10-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/hu47ZbsskEw\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=hu47ZbsskEw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\",\n  \"duration\": \"PT48M33S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw?si=MjrIlUG-8qZ-sqqw\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n<p>Defining properties of a probability refer to the rules that constitute any given probability. They are outlined below.<\/p>\n<p><!--more--><\/p>\n<p>(I) <strong>The probability of an event is always a number between 0 and 1, inclusively<\/strong>.<\/p>\n<p>$$\u00a00 \\le P(E_i) \\le 1\u00a0\\quad \\quad\u00a0 \\text{for i} = 1,2\u2026n $$<\/p>\n<p>A \u2018P\u2019 followed by E<sub>i <\/sub>in parentheses is interpreted as the probability of an event E<sub>i<\/sub>. We cannot have a negative probability or probability greater than 1 (100%). There is no such a thing as more certain than certain!<\/p>\n<p>(II)<strong> The sum of all probabilities of all events = 1, provided the events are mutually exclusive and exhaustive<\/strong>.<\/p>\n<p>$$ \\sum { P\\left( { E }_{ i } \\right) =1 } \\quad \\quad \\text{for i} = 1,2\u2026n $$<\/p>\n<p>You should note that if events are not mutually exclusive, the total probability would be greater than 1. Similarly, if events are not exhaustive, the total probability would be less than 1 (some events would be left out).<\/p>\n<h4><strong>Example<\/strong><\/h4>\n<p>Suppose we toss a <strong>fair<\/strong> coin, the only possible outcomes are either a head or a tail. Each outcome has a probability of 0.5. Therefore,<\/p>\n<p>$$ P(H) + P(T) = 0.5 + 0.5 = 1 $$<\/p>\n<p>Obtaining a head precludes obtaining a tail. Therefore, the two events are mutually exclusive. Similarly, there is no other possible outcome apart from either a head or a tail. The two events are, therefore, exhaustive.<\/p>\n<h2><strong>Empirical Probability<\/strong><\/h2>\n<p>An empirical probability is a probability that results from the analysis of actual past data. For example, if we assembled the returns earned by a stock for the last 25 years and used them to make future forecasts, then we shall, in this instance, have employed an empirical probability approach.<\/p>\n<p>One drawback about empirical probabilities is that they rely on past performance, which is not always indicative of future performance. Certain events could occur in the future, leading to drastic changes in returns.<\/p>\n<h2><strong>Subjective Probabilities<\/strong><\/h2>\n<p>Subjective probabilities usually reflect personal belief or judgment. Thus, analysts may rely on their personal experiences and judgments when projecting future performance.<\/p>\n<p>This approach is subject to personal flaws and talents. Therefore, the probabilities churned out may not be very accurate and are likely to differ, even among fund managers working for the same company.<\/p>\n<h2><strong>Priori Probabilities<\/strong><\/h2>\n<p>Priori probabilities are subjective, deductive, and based on reasoning. For example, suppose we establish that a fund manager has an 80% chance of securing a new job in a certain company, the 80% probability could have resulted either from subjective judgment or an empirical probability approach. Let\u2019s assume that the fund manager has only one competitor. If we apply deductive reasoning in this scenario, then we would conclude that the competitor has a 20% chance of securing the job.<\/p>\n\n","protected":false},"excerpt":{"rendered":"<p>Defining properties of a probability refer to the rules that constitute any given probability. They are outlined below.<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-406","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Defining Properties of Probability | CFA\u00ae Level 1 Notes<\/title>\n<meta name=\"description\" content=\"Understand the defining properties of probability in quantitative techniques. 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