Level 1 CFA® Exam:
Probability – Introduction
Let's start with the basics, which is the definition of probability and its fundamental concepts. It was commonly believed that the origins of the probability theory can be traced to European gambling in the 17th century. Most probably, however, people have always used some methods of stating probability.
Probability can be defined as the chance that an event will occur. When stating the probability of an event we can't be sure of the outcome, so the probability is a measure of the uncertainty of an event. Events that are certain will have a probability of 1, impossible ones of 0. Any value between 0 and 1 is the possible value of a probability.
Probability is often explained using the example of throwing a dice. What's the probability of rolling three dots? Intuitively, each of us would say one-sixth. Why? There are six equally likely outcomes (1, 2, 3, 4, 5, 6) and we're interested in a specific outcome, that is a 3.
A random variable is a variable that can take different numerical values depending on the case. In the case of a dice, a random variable is a number the dice may land on. When rolling a dice we don't know what number we're going to get. What's more, the number is random, which means that it depends on chance.
The set of all possible values or outcomes that a random variable can take is called a sample space. In our example, the sample space includes the outcomes 1, 2, 3, 4, 5, and 6.
An event is a subset of a sample space. An event may be one or more of the possible outcomes from a sample space.
If a set of events contains all the possible outcomes of a random variable, that is the entire sample space, we're dealing with a set of exhaustive events.
If two events can't occur at the same time, they're called mutually exclusive events. For example, a dice can't land on 2 and 3 at a time, so rolling 2 and 3 are mutually exclusive events.
2 main properties of probability:
- The probability of any event, must be a value between 0 and 1,
- The sum of the probabilities of a set of mutually exclusive and exhaustive events always equals 1.
So, it is impossible that the probability will be for example -0.5 or 1.4. Remember that if in your level 1 CFA exam you get the probability greater than 1 or lower than 0, you must have done something wrong and you must redo your calculations or rethink the whole solution.
The probability that some even A will occur can be calculated as the number of favorable outcomes divided by the sample space. For example with rolling a dice, the probability that you'll roll a 4, is always the same. The sample space will always include 6 equally likely outcomes, but we're only interested in 4 dots. The probability of rolling a 4 is therefore one-sixth. There is one favorable event: the dice landing on a 4, and there are 6 possible outcomes.
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There are also subjective probabilities which depend on individual beliefs, judgments, intuitions, and experience. A meteorologist's statement that the probability of rainfall tomorrow is 70%, which is based on his intuition, observations of the weather, and his experience, is an example of deducing a subjective probability.
Here is a couple of examples for odds for and against an event:
- If the probability of winning a bet is 1/5th >> odds for the event are 1 to 4 >> odds against the event are 4 to 1,
- If the probability of winning a bet is 2/7th >> odds for the event are 2 to 5 >> odds against the event are 5 to 2,
- If odds for winning a bet are 3 to 4 >> the probability of winning is 3/7th >> odds against winning are 4 to 3,
- If odds against winning a bet are 8 to 1 >> the probability of winning is 1/9th >> odds for winning are 1 to 8.
- Probability can be defined as the chance that an event will occur.
- A random variable is a variable that can take different numerical values depending on the case.
- The set of all possible values or outcomes that a random variable can take is called a sample space.
- An event is a subset of a sample space.
- If a set of events contains all the possible outcomes of a random variable we're dealing with a set of exhaustive events.
- If two events can't occur at the same time, they're called mutually exclusive events.
- Two main properties of probability: (A) the probability of any event, must be a value between 0 and 1, (B) the sum of the probabilities of a set of mutually exclusive and exhaustive events always equals 1.
- In the case of a priori probability we assume that the occurrence of each event is equally likely and the probability can be determined before the event occurs.
- Empirical probability is a probability based on the results of an experiment.
- A proriori probability and empirical probability are examples of objective probability.
- Subjective probabilities depend on individual beliefs, judgments, intuitions, and experience.