Introduction to Probability

Subjective Probability

A type of probability based on personal beliefs, judgment, or experience about the occurrence of a specific outcome in the future. The calculation of subjective probability contains no formal computations (of any formula) and reflects the opinion of a person based on his/her experience. The subjective probability differs from subject to subject and it may contain a high degree of personal biases.

This kind of probability is usually based on a person’s experience, understanding, knowledge, and intelligence and determines the probability of some specific event (situation). It is usually applied in real-life situations, especially, related to the decision in business, job interviews, promotions of the employee, awarding incentives, and daily life situations such as buying and/or selling of a product. An individual may use their expertise, opinion, past experiences, or intuition to assign the degrees of probability to a specific situation.

It is worth noting that the subjective probability is highly flexible in terms of an individual’s belief, for example, one individual may believe that the chance of occurrence of a certain event is 25%. The same person or others may have a different belief especially when they are given a specific range from which to choose, (such as 25% to 30%). This can occur even if no additional hard data is behind the change.

Events that may Alter Subjective Probability

Subjective probability is usually affected by a variety of personal beliefs and opinions (related to his caste, family, region, religion, and even relationship with people, etc.), held by an individual. It is because the subjective probability is often based on how each individual interprets the information presented to him

Disadvantages of Subjective Probability

As only personal opinions (beliefs, experiences) are involved, there may be a high degree of bias. On the other hand, one person’s opinion may differ greatly from the opinion of another person. Similarly, in subjective probability, one may fall into the trap of failing to meet complex calculations.

Subjective Probability

Examples Related to Subjective Probability

  • One may think that there is an 80% chance that your best friend will call you today because his/her car broke down yesterday and he/she will probably need a ride.
  • You think you have a 50% chance of getting a certain job you applied for as the other applicant is also qualified.
  • The probability that in the next (say) 5 hours, there will be rain is based on current weather situations, wind patterns, nearby weather, barometric pressure, etc. One can predict this based on his experience with weather and rain, and believes, in predicting the rain in the next 5 hours.
  • Suppose, a cricket tournament is going to be held between Pakistan and India. The theoretical probability of winning either the cricket team is 50%. But, in reality, it is not 50%. On the other hand (like empirical probability), the number of trial tournaments cannot be arranged to determine an experimental probability. Thus, the subjective probability will be used to find the winning team which will be based on the beliefs and experience of the investigator who is interested in finding the probability of the Pakistan cricket team as the winner. Note there will be a bias if any of the fans of a team investigates the probability of winning a team.
  • To locate petroleum, minerals, and/ or water lying under the earth, dowsers are employed to predict the likelihood of the existence of the required material. They usually adopt some non-scientific methods. In such a situation, the subject probability is used.
  • Note the decisions through subjective probability may be valid if the degree of belief of a person is unbiased about the situation and he/she arrives by some logical reasoning.

For further reading See Introduction to Probability

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Classical Probability: Example, Definition, and Uses in Life

Classical probability is the statistical concept that measures the likelihood (probability) of something happening. In a classic sense, it means that every statistical experiment will contain elements that are equally likely to happen (equal chances of occurrence of something). Therefore, the concept of classical probability is the simplest form of probability that has equal odds of something happening.

Classical Probability Examples

Example 1: The typical example of classical probability would be rolling a fair die because it is equally probable that the top face of the die will be any of the 6 numbers on the die: 1, 2, 3, 4, 5, or 6.

Example 2: Another example of classical probability would be tossing an unbiased coin. There is an equal probability that your toss will yield either head or tail.

Example 3: In selecting bingo balls, each numbered ball has an equal chance of being chosen.

Example 4: Guessing a multiple choice quiz (MCQs) test with (say) four possible answers A, B, C, or D. Each option (choice) has the same odds (equal chances) of being picked (assuming you pick randomly and do not follow any pattern).

Classical Probability Formula

The probability of a simple event happening is the number of times the event can happen, divided by the number of possible events (outcomes).

Mathematically $P(A) = \frac{f}{N}$,

where, $P(A)$ means “probability of event A” (event $A$ is whatever event you are looking for, like winning the lottery, that is event of interest), $f$ is the frequency, or number of possible times the event could happen and $N$ is the number of times the event could happen.

For example,  the odds of rolling a 2 on a fair die are one out of 6, (1/6). In other words, one possible outcome (there is only one way to roll a 1 on a fair die) is divided by the number of possible outcomes.

Classical probability can be used for very basic events, like rolling a dice and tossing a coin, it can also be used when the occurrence of all events is equally likely. Choosing a card from a standard deck of cards gives you a 1/52 chance of getting a particular card, no matter what card you choose. On the other hand, figuring out whether will it rain tomorrow or not isn’t something you can figure out with this basic type of probability. There might be a 15% chance of rain (and therefore, an 85% chance of it not raining).

Classical Probability Formula

Other Examples of classical Probability

There are many other examples of classical probability problems besides rolling dice. These examples include flipping coins, drawing cards from a deck, guessing on a multiple-choice test, selecting jellybeans from a bag, choosing people for a committee, etc.

Classical Probability cannot be used:

Dividing the number of events by the number of possible events is very simplistic, and it isn’t suited to finding probabilities for a lot of situations. For example, natural events like weights, heights, and test scores need normal distribution probability charts to calculate probabilities. Most “real life” things aren’t simple events like coins, cards, or dice. You’ll need something more complicated than classical probability theory to solve them.

For further Details see Introduction to Probability

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Online MCQs Quiz Website of Various Subjects

Probability Theory: An Introduction

This post is about probability theory. It will serve as an introduction to the theory of chances.

Probability Theory

Uncertainty is everywhere i.e. nothing in this world is perfect or 100% certain except the Almighty Allah the Creator of the Universe. For example, if someone bought 10 lottery tickets out of 500 and each of the 500 tickets is as likely as any other to be selected or drawn for the first prize then it means that you have 10 chances out of 500 tickets or 2% chances to win the first prize.

Similarly, a decision maker seldom has complete information to make a decision.
So, probability is a measure of the likelihood that something will happen, however, probability cannot predict the number of times that something will occur in the future, so all the known risks involved must be scientifically evaluated. The decisions that affect our daily life, are based upon the likelihood (probability or chance) but not on absolute certainty. The use of probability theory allows the decision maker with only limited information to analyze the risks and minimize the gamble inherently. For example in marketing a new product or accepting an incoming shipment possibly containing defective parts.

Probability Theory

Probability can be considered as the quantification of uncertainty or likelihood. Probabilities are usually expressed as fractions such as {1/6, 1/2, 8/9} or as decimals such as {0.167, 0.5, 0.889} and can also be presented as percentages such as {16.7%, 50%, 88.9%}.

Types of Probability

Suppose we want to compute the chances (Note that we are not predicting here, just measuring the chances) that something will occur in the future. For this purpose, we have three types of probability

1) Classical Approach or Prior Approach

In a classical probability approach, two assumptions are used

Classical probability is defined as “The number of outcomes favorable to the occurrence of an event divided by the total number of all possible outcomes”.
OR
An experiment resulting $n$ equally likely mutually exclusive and collectively exhaustive outcomes and “m” of which are favorable to the occurrence of an event A, then the probability of event A is the ratio of m/n. (D.S. Laplace (1749-1927).

Symbolically we can write $P(A) = \frac{m}{n} = \frac{number\,\, of\,\, favorable\,\, outcomes}{Total\,\, number\,\, of\,\, outcomes}$

Some shortcomings of the classical approach

  • This approach to probability is useful only when one deals with card games, dice games, or coin tosses. i.e. Events are equally likely but not suitable for serious problems such as decisions in management.
  • This approach assumes a world that does not exist, as some assumptions are imposed as described above.
  • This approach assumes symmetry about the world but there may be some disorder in a system.

2) Relative Frequency or Empirical Probability or A Posterior Approach

The proportion of times that an event occurs in the long run when conditions are stable. Relative frequency becomes stable as the number of trials becomes large under uniform conditions.
To calculate the relative frequency an experiment is repeated a large number of times say “n” under uniform/stable conditions. So if an event A occurs m times, then the probability of the occurrence of the event A is defined by
$P(A)=\lim_{x\to\infty}\frac{m}{n}$

if we say that the probability of a number n child will be a boy is 1/2, then it means that over a large number of children born 50% of all will be boys.

Some Critics

  • It is difficult to ensure that the experiment is repeated under stable/uniform conditions.
  • The experiment can be repeated only a finite number of times in the real world, not an infinite number of times.

3) Subjective Approach

This is the probability based on the beliefs of the persons making the probability assessment.
Subjective probability assessments are often found when events occur only once or at most a very few times.
This approach is applicable in business, marketing, and economics for quick decisions without performing any mathematical calculations.
The Disadvantage of subjective probability is that two or more persons facing the same evidence/problem may arrive at different probabilities i.e. for the same problem there may be different decisions.

Real-Life Example of Subjective Probability

  • A firm must decide whether or not to market a new type of product. The decision will be based on prior information that the product will have high market acceptance.
  • The Sales Manager considers that there are 40% chance of obtaining the order for which the firm has just quoted. This value (40% chance) cannot be tested by repeated trials.
  • Estimating the probability that you will be married before the age of 30 years.
  • Estimating the likelihood (probability, chances) that Pakistan’s budget deficit will be reduced by half in the next 5 years.

Note that subjective probability theory is not a repeatable experiment, the relative frequency approach to probability is not applicable, nor can equally likely probabilities be assigned.

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