Understanding Probability Concepts | CFA Level I Quantitative Methods
Probability concepts are essential to investors, as they allow them to make informed decisions based on past data and future predictions. In this article, we’ll explore some fundamental definitions in probability concepts, including random variables, events, probabilities, and odds. So, let’s get started!
Common Definitions in Probability
Random variable: An uncertain quantity or number, such as the monthly return of an asset. Random variables vary over time, making them uncertain and hence “random.” We usually denote random variables with capital letters, e.g., X for the monthly return of asset A.
Outcome: The observed value of a random variable, like the monthly return values. For an investor, each observed monthly return is an outcome, which is a specific instance of the random variable.
Event: A single outcome or a set of outcomes. For example, X=0 (no profit, no loss) or X ≤ 0 (negative or zero returns). Events are also denoted by capital letters, such as A. Events can be simple, like a single outcome, or complex, comprising multiple outcomes.
Mutually exclusive events: Events that cannot both happen at the same time. If event B represents the monthly return being positive, events A and B are mutually exclusive since a monthly return cannot be both positive and negative simultaneously.
Exhaustive events: A set of events that include all possible outcomes. In our example, events A and B are exhaustive and mutually exclusive. However, exhaustive events do not have to be mutually exclusive, as they can sometimes overlap.
Defining Properties of Probability
Probability has two main defining properties that help us understand and work with these concepts:
Sum of probabilities: If a set of events is both mutually exclusive and exhaustive, the sum of their probabilities always equals 1. In other words, the total probability of all possible outcomes is 100%. Range of probabilities: The probability of any event is between 0 and 1. A probability of 0 means the event will never happen, while a probability of 1 means the event is certain to occur. This range is essential in understanding the likelihood of different events happening.
Determining Probability
As an analyst, you can use various methods to determine the probability of an event:
Empirical method: This method involves analyzing past data and making inferences based on historical patterns. By examining previous occurrences, you can make predictions about the likelihood of future events.
Subjective method: Basing probability on personal judgment and experience is the least formal method. It’s called the “subjective method” because it depends on an individual’s perspective and intuition.
A priori probability: Using formal reasoning based on logical explanations, this method relies on inherent knowledge or understanding of a situation.
A priori and empirical probabilities are generally considered objective probabilities because they do not vary from person to person.
Probability and Odds
Probability can be stated as odds, which is another way of expressing the likelihood of an event happening.
Odds for = Probability of event / (1 – Probability of event)
EXAMPLE
Calculate the odds for event E with a probability of 0.2:
Odds for = Probability of event / (1 – Probability of event)
Odds for = 0.2 / (1 – 0.2) = 0.2 / 0.8 = 1 / 4
The odds for event E are 1 to 4, which means we expect 1 occurrence of E for every 4 cases of non-occurrence.
Odds against: The reciprocal of odds for, odds against are more commonly used in betting. For example, with odds against of “4 to 1”, you will receive a $4 profit for every $1 wagered, plus your original $1 back. This means that you will receive a total of $5 for every $1 wagered!
Conclusion
Understanding probability concepts is crucial for making informed decisions in the world of finance. By familiarizing yourself with common definitions, properties of probability, and methods to determine probabilities, you’re building a strong foundation for tackling probability problems. In the next lesson, we’ll delve deeper into probability tools and techniques that will further enhance your knowledge. See you soon!
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