Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Category: Introduction to Probability

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 of a fair dice because it is equally probable that top face of 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).

Formula for Classical Probability

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) 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 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 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).

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, and 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. In fact, 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 Detail see Introduction to Probability

Introduction Probability Theory

Introduction Probability Theory

Uncertainty is every where 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 500 tickets is as likely as any other to be selected or drawn for first prize then it means that you have 10 chances out of 500 tickets or 2% chances to win a first prize.

Similarly, a decision maker seldom have the complete information to make a decision.
So probability is a measure of likelihood that something will happen, however probability cannot predict the number of times that something will occur in future, so it is important that all the known risks involved 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 inherent. For example in marketing a new product or accepting an incoming shipment possibly containing defective parts.

Probability can be considered as the quantification of uncertainty or the likelihood. Probabilities are usually expressed as fraction 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 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”.
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 ration 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 shortcoming of classical approach

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

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 the stable/uniform conditions.
  • Experiment can be repeated only a finite number of times in 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, 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 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 prior information that the product will have high market acceptance.
  • The Sales Manager considers that there is 40% chances of obtaining the order for which the firm has just quoted. This value (40% chances) 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 budget deficit will be reduced by half in the next 5 years.

Note that subjective probability 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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