Introduction To Probability

• A probability measures the likelihood that some event occurs.
• In most financial applications of probability, events are tightly coupled with numeric values. Examples of such events include
  • the loss on a portfolio,
  • the number of defaults in a mortgage pool, or
  • the sensitivity of a portfolio to a rise in short-term interest rates.
• Events can also be measured by values without a natural numeric correspondence. Examples include categorical variables, such as
  • the type of a financial institution, or
  • the rating on a corporate bond.

Probability – Basic Definitions

• Random Experiment → It is an experiment whose result is unknown.
• Random Variable → It is a variable whose value is unknown.
• Outcome → It is the observed value of a random variable.
• Sample Space → It is the set of all possible outcomes.
• Event → It is a subset of the sample space i.e. one or several outcomes out of it.
• One of the mathematical definitions of Probability of an event to occur can be –
  • Number of outcomes favorable for the event, over the number of total possible outcomes in a random experiment.

Sample Space

• A sample space is a set containing all possible outcomes of an experiment and is denoted as Ω. The set of outcomes depends on the problem being studied.
• The event space, usually denoted by F, consists of all combinations of outcomes to which probabilities can be assigned.

Event Space

• The event space, usually denoted by F, consists of all combinations of outcomes to which probabilities can be assigned.
• A definite probability of 0 can be assigned to “impossible” events.

Interpretation Of Probability

• Probability measures the likelihood of an event.
• The simplest interpretation of probability is that it is the frequency with which an event would occur if a set of independent experiments was run. This interpretation of probability is known as the frequentist interpretation and focuses on objective probability.
• Probability can also be interpreted from a subjective point of view. Under this alternative interpretation, probability reflects or incorporates an individual’s beliefs about the likelihood of an event occurring. These beliefs may differ across individuals and do not have to agree with the objective probability of an event.
• Probability is defined over event spaces. Mathematically, it assigns a number between 0 and 1 (inclusive) to each event in the event space.

Mutually Exclusive Events

• Mutually exclusive events are events which cannot happen together i.e. P(A∩B)=0 if A and B are mutually exclusive events. e.g. In a toss of a coin, the result can either be heads or tails but not both.
• In general
• But if A and B are mutually exclusive events.
• Hence, for a given random variable, the probability of any of two mutually exclusive events occurring is just the sum of their individual probabilities i.e.

  Example

  Suppose A is the event that stock return is less than 5%, and B is the event that stock return is greater than 10%. It is given that the probability that either of the two will happen is 0.65, and the probability that the stock return will be greater than or equal to 5% is also 0.65. Find the probability of event B happening i.e. find the probability that the stock return is greater than 10%.

Fundamental Principles Of Probability

• The three fundamental principles of probability are defined using events, event spaces, and sample spaces. Note that is a function that returns the probability of an event .
• Any event A in the event space has .
• The probability of all events in is one and thus .
• If the events are mutually exclusive, then . This holds for any number n of mutually exclusive events, so that

• These three principles, collectively known as the Axioms of Probability—are the foundations of probability theory.

Exhaustive Events

• When two or more events form the sample space, they are known as exhaustive events. In that case, there is a certain chance of occurrence of at least one of them when they are all considered together.

Independent Events

• Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring i.e. the occurrence of A does not influence (and is not influenced by) the occurrence of B. e.g. if a die is rolled twice, the event of 5 occurring in the next throw won’t influence, and won’t be influenced by the occurrence of 5 in the first throw.
• For independent events A and B, the probability that both will happen is simply the product of the probabilities of the two events occurring individually i.e.
• Hence for independent events
• The probability of two events occurring together is referred to as joint probability.

  Example

  Suppose A is the event that a head is obtained when a coin is tossed and B is the event that a 5 is obtained when a single 6-sided die is rolled. Find the probability of getting a head when the coin is tossed and obtaining a number other than 5 when the die is rolled.

Unconditional Probability

• Unconditional probability (or marginal probability) is the probability of an event irrespective of the past or future occurrence of any other event. For example, the probability of the stock outperforming the market, irrespective of the economy expanding or receding, is the unconditional probability of the stock outperforming. For any event A, it is calculated as :

where n(A) is the number of favorable outcomes for event A and n(S) is the total number of possible outcomes or number of outcomes in sample space S.

Conditional Probability

• Conditional probability is the probability of an event given that another event has occurred. For Example, the probability of the stock outperforming the market, given that the economy is expanding, is conditional probability for stock outperforming.
• It is denoted as P(A|B) which means the conditional probability of event A occurring given that event B has already occurred.

If two events A and B are independent then

P(A│B)=P(A)

Joint Probability Algebra

• Joint probability of two events is the probability that both will occur together, and can only be calculated where more than one observation can occur at the same time. Joint probability of two events A and B is denoted as P(A∩B) or P(AB) and is calculated as:

  

  where P(A|B) is the conditional probability of event A occurring given that event B has already occurred and P(B) is the unconditional probability of event B.

Joint Probability Algebra – Example

• The probability that the economy will undergo a slowdown is 0.5 and the probability that the Fed will increase the interest rates is 0.6. The probability that the economy will undergo a slowdown given that the Fed will increase the interest rates is 0.7. Find the probability that:
  • Slowdown will happen and interest rates will be increased
  • Interest rates will increase given that slowdown will happen
  • At least one of the two events will happen

Conditional Independence

• Like probability, independence can be redefined to hold conditional on another event. Two events A and B are conditionally independent if:

Theorem Of Total Probability

• Unconditional probability can be calculated from conditional probability if the events which have already occurred are mutually exclusive and exhaustive (i.e. they cannot occur together and their probabilities add to 1)

Given n mutually exclusive events whose probabilities sum to 1, then the unconditional probability of event B

• The unconditional probability of event A is equal to the sum of probabilities of the following two events:

where is the probability of not happening of A

Theorem Of Total Probability – Example

• Out of a group of 1000 patients being treated for chronic back trouble, 30% are chosen at random to receive a new, experimental treatment as opposed to the more usual muscle relaxant-based therapy which the remaining patients receive. Preliminary studies suggest that the probability of a cure with the standard treatment is 0.2, while the probability of a cure from the new treatment is 0.6. How many patients (on average) out of the 100 patients selected at random would be cured?

Bayes’ Theorem

• For two random variables, A and B :

• Bayes’ Theorem is a plan to change our beliefs in the face of new evidence.

• Even though the expression of Bayes’ Theorem is very simple, yet its application can be quite complex. Bayesian analysis is used heavily in the fields of finance and risk management.

Bayes’ Theorem – Example 1

Consider our last example again. Out of a group of 100 patients being treated for chronic back trouble, 30% are chosen at random to receive a new, experimental treatment as opposed to the more usual muscle relaxant-based therapy which the remaining patients receive. Preliminary studies suggest that the probability of a cure with the standard treatment is 0.2, while the probability of a cure from the new treatment is 0.6. Some time later, one of the patients returns to thank the staff for her complete recovery. What is the probability that she was given the new treatment?

Bayes’ Theorem – Example 2

A factory has three machines A, B and C, which produce 200, 400 and 500 bulbs daily. The machines A, B and C produce 1%, 3% and 5% defective bulbs respectively. During testing, a bulb is picked randomly and is found to be defective. Find the probability that is produced by machine A.

Bayes’ Theorem – Example 3

You are an analyst at Astra Fund of Funds. Based on an examination of historical data, you determine that all fund managers fall into one of two groups. Stars are the best managers. The probability that a star will beat the market in any given year is 75%. Other managers are just as likely to beat the market as they are to underperform it [i.e., non-stars have 50/50 odds of beating, P[B|S] = 50%]. For both types of managers, the probability of beating the market is independent from one year to the next. Stars are rare. Of a given pool of managers, only 16% turn out to be stars. A new manager was added to your portfolio of funds three years ago. Since then, the new manager has beaten the market every year. What was the probability that the manager was a star when the manager was first added to the portfolio? What is the probability that this manager is a star now?