Solutions to Assignments
MBA and MBA (Banking & Finance)
MMPC-005 - Quantitative Analysis for Managerial
Applications
Question No. 2.
Explain the concept of probability theory. Also, explain what are the different approaches to probability theory.
probability theory, a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory of probability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples. The distinctive feature of games of chance is that the outcome of a given trial cannot be predicted with certainty, although the collective results of a large number of trials display some regularity. For example, the statement that the probability of “heads” in tossing a coin equals one-half, according to the relative frequency interpretation, implies that in a large number of tosses the relative frequency with which “heads” actually occurs will be approximately one-half, although it contains no implication concerning the outcome of any given toss. There are many similar examples involving groups of people, molecules of a gas, genes, and so on. Actuarial statements about the life expectancy for persons of a certain age describe the collective experience of a large number of individuals but do not purport to say what will happen to any particular person. Similarly, predictions about the chance of a genetic disease occurring in a child of parents having a known genetic makeup are statements about relative frequencies of occurrence in a large number of cases but are not predictions about a given individual.
The fundamental ingredient of probability theory is an experiment that can be repeated, at least hypothetically, under essentially identical conditions and that may lead to different outcomes on different trials. The set of all possible outcomes of an experiment is called a “sample space.” The experiment of tossing a coin once results in a sample space with two possible outcomes, “heads” and “tails.” Tossing two dice has a sample space with 36 possible outcomes, each of which can be identified with an ordered pair (i, j), where i and j assume one of the values 1, 2, 3, 4, 5, 6 and denote the faces showing on the individual dice. It is important to think of the dice as identifiable (say by a difference in colour), so that the outcome (1, 2) is different from (2, 1). An “event” is a well-defined subset of the sample space. For example, the event “the sum of the faces showing on the two dice equals six” consists of the five outcomes (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).
A third example is to draw n balls from an urn containing balls of various colours. A generic outcome to this experiment is an n-tuple, where the ith entry specifies the colour of the ball obtained on the ith draw (i = 1, 2,…, n). In spite of the simplicity of this experiment, a thorough understanding gives the theoretical basis for opinion polls and sample surveys. For example, individuals in a population favouring a particular candidate in an election may be identified with balls of a particular colour, those favouring a different candidate may be identified with a different colour, and so on. Probability theory provides the basis for learning about the contents of the urn from the sample of balls drawn from the urn; an application is to learn about the electoral preferences of a population on the basis of a sample drawn from that population.
Another application of simple urn models is to use clinical trials designed to determine whether a new treatment for a disease, a new drug, or a new surgical procedure is better than a standard treatment. In the simple case in which treatment can be regarded as either success or failure, the goal of the clinical trial is to discover whether the new treatment more frequently leads to success than does the standard treatment. Patients with the disease can be identified with balls in an urn. The red balls are those patients who are cured by the new treatment, and the black balls are those not cured. Usually there is a control group, who receive the standard treatment. They are represented by a second urn with a possibly different fraction of red balls. The goal of the experiment of drawing some number of balls from each urn is to discover on the basis of the sample which urn has the larger fraction of red balls. A variation of this idea can be used to test the efficacy of a new vaccine. Perhaps the largest and most famous example was the test of the Salk vaccine for poliomyelitis conducted in 1954. It was organized by the U.S. Public Health Service and involved almost two million children. Its success has led to the almost complete elimination of polio as a health problem in the industrialized parts of the world. Strictly speaking, these applications are problems of statistics, for which the foundations are provided by probability theory.
In contrast to the experiments described above, many experiments have infinitely many possible outcomes. For example, one can toss a coin until “heads” appears for the first time. The number of possible tosses is n = 1, 2,…. Another example is to twirl a spinner. For an idealized spinner made from a straight line segment having no width and pivoted at its centre, the set of possible outcomes is the set of all angles that the final position of the spinner makes with some fixed direction, equivalently all real numbers in [0, 2π). Many measurements in the natural and social sciences, such as volume, voltage, temperature, reaction time, marginal income, and so on, are made on continuous scales and at least in theory involve infinitely many possible values. If the repeated measurements on different subjects or at different times on the same subject can lead to different outcomes, probability theory is a possible tool to study this variability.
Because of their comparative simplicity, experiments with finite sample spaces are discussed first. In the early development of probability theory, mathematicians considered only those experiments for which it seemed reasonable, based on considerations of symmetry, to suppose that all outcomes of the experiment were “equally likely.” Then in a large number of trials all outcomes should occur with approximately the same frequency. The probability of an event is defined to be the ratio of the number of cases favourable to the event—i.e., the number of outcomes in the subset of the sample space defining the event—to the total number of cases. Thus, the 36 possible outcomes in the throw of two dice are assumed equally likely, and the probability of obtaining “six” is the number of favourable cases, 5, divided by 36, or 5/36.
Approaches
Classical or Mathematical Definition of Probability
Let’s say that an experiment can result in (m + n), equally likely, mutually exclusive, and exhaustive cases. Also, ‘m’ cases are favorable to the occurrence of an event ‘A’ and the remaining ‘n’ are against it. In such cases, the definition of the probability of occurrence of the event ‘A’ is the following ratio:
𝑚𝑚+𝑛 = \( \frac {\text {Number of cases favorable to the occurrence of the event ‘A’}}{\text {Total number of equally likely, mutually exclusive, and exhaustive cases}} \)
The probability of the occurrence of the event ‘A’ is P(A). Further, P(A) always lies between 0 and 1. These are the limits of probability.
Instead of saying that the probability of the occurrence of the event ‘A’ is 𝑚𝑚+𝑛, we can say that “Odds are m to n in favor of event A or n to m against the event A.” Therefore,
Odds in favor of the event A = No. of cases favorable to the occurrence of the event ANo. of cases against the occurrence of the event A = 𝑚𝑚+𝑛𝑛𝑚+𝑛 = 𝑚𝑛
Odds against the event A = Number of cases against the occurrence of the event ANumber of cases in favour of the occurrence of the event A = 𝑛𝑚
Note: The ratio 𝑚𝑛 or 𝑛𝑚 is always expressed in its lowest form (integers with no common factors).
- If m = 0, or if the number of cases favorable to the occurrence of the event A = 0 then, P(A) = 0. In other words, event A is an impossible event.
- If n = m, then P(A) = 𝑚𝑚 = 1. This means that the event A is a certain or sure event.
- If neither m = 0 nor n = 0, then the probability of occurrence of any event A is always less than 1. Therefore, the probability of occurrence the event satisfies the relation 0 < P < 1.
If the events are mutually exclusive and exhaustive, then the sum of their individual probabilities of occurrence = 1.
For example, if A, B, and C are three mutually exclusive events, then P(A) + P(B) + P(C) = 1. The probability of the occurrence of one particular event is the Marginal Probability of that event.
Choosing an object at random from N objects means that each object has the same probability 1𝑁 of being chosen.
Source: Wikipedia
Empirical Probability or Relative Frequency Probability Theory
The Relative Frequency Probability Theory is as follows:
We can define the probability of an event as the relative frequency with which it occurs in an indefinitely large number of trials. Therefore, if an event occurs ‘a’ times out of ‘n’, then its relative frequency is 𝑎𝑛.
Further, the value that 𝑎𝑛 approaches when ‘n’ becomes infinity is the limit of the relative frequency.
Symbolically,
P(A) = lim𝑛→∞𝑎𝑛
However, in practice, we write the estimate of P(A) as follows:
P(A) = 𝑎𝑛
While the classical probability is normally encountered in problems dealing with games of chance. On the other hand, the empirical probability is the probability derived from past experience and is used in many practical problems.
Total Probability Theorem or the Addition Rule of Probability
If A and B are two events, then the probability that at least one of then occurs is P(A∪B). We also have,
P(A∪B) = P(A) + P(B) – P(A∩B)
If the two events are mutually exclusive, then P(A∩B) = 0. In such cases, P(A∪B) = P(A) + P(B).
Multiplication Rule
If A and B are two events, the probability of their joint or simultaneous occurrence is:
P(A∩B) = P(A) . P(A/B)
If the events are independent, then
- P(A/B) = P(A)
- P(B/A) = P(B)
Therefore, we now have,
If the events are independent, then P(A∪B) = P(A) + P(B) – P(A∩B)
Also, P(A/B) is the conditional probability of the occurrence of the event A when event B has already occurred. Similarly, P(B/A) is the conditional probability of the occurrence of event B when event A has already occurred. If the events are independent, then the occurrence of A does not affect the occurrence of B.
∴ P(B/A) = P(B)
Also, P(A/B) = P(A)
