Uncertainty is all around us and we often come across real-life situations when we have to decide on making a choice from the available options. Questions like “Will it rain? Do I need to carry an umbrella today?” or “Will there be a rise in taxes? Which party will win the election this time?” All these situations demand a decision from us and this is the time when probability theory comes to our rescue. From weather forecasts, opinion polls to making business decisions, the concepts of probability come in handy at various aspects of our daily lives. What are the chances? Learn how you can actually predict the chances, with probability analysis in this course.
Whether you are an economist, a businessman or a manager, you will come across instances when you have to face uncertainty with respect to the outcomes of your business decisions. For example, when you have to launch a new product into the market, you will need to weigh in factors like market demand, customer perception and usefulness of the product in the targeted area. Probability theory helps managers and businessmen to select the right markets and the best time to launch the product based on prior surveys and customer information etc. You can learn probability concepts, techniques and decision making with this course.
What is Theoretical Probability?
While probability theory focuses on the likelihood of an event taking place, theoretical probability is all about the occurrence of an event based on all possible outcomes that are already known. To simply explain this statement, take for instance the tossing of a coin. We know that the result would be either head or tail, which are equally likely. By means of theoretical probability, we know that the likelihood of a head or tail coming on top is same i.e. ½ or 0.5. If you are looking to build a solid foundation towards understanding probability theory, take up this course.
Most real-life events can be predicted based on their occurrences in the past by studying the data collected from surveys, trials, opinion polls etc. Such kind of probability theory is called Empirical Probability and it’s based on observations and experiences. For example, if we have to find the probability that a child will choose vanilla ice cream from a range of flavors, then we need the data from a survey. Suppose from a group of 100 children, 35 chose vanilla, then the probability that a child will prefer vanilla flavor over others is 35/100 i.e. 0.35.
Theoretical probability on the other hand uses the knowledge on the likely outcomes to find the different ways in which the event can occur. It is the method of finding the probability of an event from a sample space of known and equally likely outcomes. This simple means that each outcome is as likely to occur as the other. To explain this with an example, take for instance the rolling of a fair die. How do you find the probability of rolling a ‘six’? You first need to determine the sample space of equally likely events. In this case, it’s (1, 2, 3, 4, 5, and 6). The theoretical probability of rolling a six is 1 out of 6, i.e. 1/6. The probability of rolling six will be represented as P(E), where P is the theoretical probability and E is the event in consideration.
Therefore, we can say that theoretical probability uses analytical knowledge on the probable outcomes to determine the probability of an event instead of using experimentation. Learn more about probability theory and its implementation in real-life scenarios with this workshop on Probability.
Rules of Probability
Before we understand the basic rules of probability, let’s have a look at some of the common definitions that you must know:
- Experiment – Any uncertain process that’s under study is called an experiment. In our example of tossing a coin, the toss is the experiment.
- Outcome – The result of an experiment is called outcome. The occurrence of ‘head’ on top in the tossing of a coin is an example of an outcome.
- Sample Space – The set of all possible outcomes is called Sample Space. In our example, since head and tail are the two possible outcomes, the sample space would be denoted as S = (H, T)
- Size of Sample Space – The total number of outcomes possible in an experiment is called the size of the sample space and here it would be denoted as n(S) = 2 since there are only two outcomes in a tossing of a coin.
- Event – The event is defined as the specific outcome of an experiment that you might be interested in. Suppose while tossing the coin, you want heads up, then the event will be denoted as E = (H) with a sample size n(E) = 1
- Probability – It is the likelihood of the occurrence of an event and is a number between 0 and 1. If probability is 0, it means that the event can never occur and if its 1, it means that the event will always occur. In case of a fraction like ½ i.e. 0.5, it means that the event will occur 1 out of two times. For an in-depth knowledge on the different kinds of probability theorems, visit our advanced course on Probability.
- Theoretical Probability – When the possible outcomes of an event have an equal chance of occurring, then it’s called a theoretical probability. It is defined as the ratio of ‘number of outcomes in the event set’ to the ‘number of possible outcomes in the sample space’ or simply put P (E) = n (E) / n(S). In our coin tossing example, P(E) = ½ with E being the event of the coin landing heads up.
Statistical Independence and Dependence
Statistically independent events are the ones that have no bearing whatsoever on the probability of occurrence of another event. For example, the gender of the second child in a family is statistically independent of the gender of the older child in the family.
Statistically dependent events, on the other hand, are the events that have an effect on the probability of occurrence of other events. For instance, let’s say we have 10 different flavoured ice cream cones with only 1 being a vanilla flavoured cone. When a kid, who is not particular about vanilla, is asked to choose an ice cream cone first, then the probability that the next kid, who prefers vanilla ice cream, actually gets his choice depends on the first kid’s selection. Here, the events are statistically dependent.
Marginal, Joint and Conditional Probabilities
Marginal Probability is the probability of the occurrence of a particular event only. In our example of the gender of the second child in a family, the event that it’s a male child i.e. P (male child) = 0.5 because it could either be a boy or a girl. Get more insights into the different concepts of statistics in our introductory Course on Statistics.
Joint Probability is the probability of two or more events happening together or one after another. For example, when two dice are rolled at the same time; the probability that a six will appear on both dice will be represented as Joint Probability, P (A, B) = 1/6 x 1/6, which is 0.02777.
Conditional Probability refers to the probability that a second event B will occur given that a first event A has already occurred. For example, what is the probability that a person will buy a soft drink when he has already purchased a bottle of wine? It’s represented as P (B|A) with B being the event of buying soft drink and A being the event that the shopper bought a wine bottle.
These probability theories are used by business analysts and statisticians to deduce the relationship between simultaneous events happening at a given time. To give an example, when the industrial share prices fall, the value of the dollar also declines. Analysts use probability theory for a number of applications right from business, economy to politics and education.
Importance of Statistics
While theoretical probability is based on the prior knowledge on the possible outcomes, in some cases it’s difficult to compute the theoretical probability of an event. For example, how do we know that baseball team A will win this season? The probability depends on their past record, player performance and other factors. We need to look into the historical data to arrive at a probability; the more the team’s success rate, the better its chances of winning the title. For this reason, statistics and statistical analysis is very important in deducing the probability of complex events. Check out this course in Big Data analysis that shows how to use statistical concepts to work with Big Data.
Very often managers, scientists, political leaders and businessmen etc. have to make critical decisions based on the occurrence of an event or the uncertainty that surrounds it. If you are the owner of a small business or a medium enterprise, then you will find the various tools and techniques of theoretical probability very useful for making important decisions. In short, the relevance of probability theory is not limited to statisticians and analysts; it helps decision-makers from all walks of life to deal with uncertainties in an informed way.