PROBABILITY

People are always interested in their future, and what will happen tomorrow is sometimes more important to us than what is happening now. Tomorrow is like compensation and a nice revenge for us. Most people are fascinated by tomorrow - What would our life look like? Am I going to be richer and prettier? It is like an escape from today. Because today is always guilty for something. That’s why we often try to predict the future. Sometimes our predictions are based on what we know and we have some choices to choose between. In this case we just use some techniques and methods to give more chances to some possible events to others. This kind of predictability is an object of Mathematics. When our predictions are not based on real fact it is hard to decide which event has more chances to occur in the future. In this case it is like playing blind game with no certain rules.

In this short text it won’t be possible to describe everything that relates somehow to probability. We will try to open the door so as to allow the curiosity to come in.

There are many situations in real life where we have to take a chance or risk. Based on certain situations, the chance of occurrence of a certain event can be easily predicted. In simple words, the chance of occurrence of a particular event is what we study in probability. 

Probability is a branch of mathematics concerning how likely something is to happen. The most simple definition of probability is that probability is a number between 0 and 1, where 0 indicates that something is impossible to happen, and 1 indicates that something is certain to happen. A simple example that illustrates this idea is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

When dealing with experiments that are random and well-defined  in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes. For example, tossing a fair coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. 

Many events cannot be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The probability of any one of them is ⅙.

Probability is just a guide. It does not tell us exactly what will happen. Example: toss a coin 100 times, how many Heads will come up? Probability says that heads have a ½ chance, so we can expect 50 Heads. But when we actually try we might get 42, or 52 heads, or anything really, but in most cases it will be a number near 50.

Some words have special meaning in Probability: Experiment is a repeatable procedure with a set of possible results. Example: Throwing a die. We can throw the die again and again, so it is repeatable. The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}. Outcome is a possible result in an experiment. Example: Getting a "6". Sample space is all the possible outcomes of an experiment. Example: Choosing a card from a deck. There are 52 cards on the desk. So the Sample space is all 52 possible cards. The Sample Space is made up of Sample Points: Sample point is just one of the possible outcomes. Event is one or more outcomes of an experiment.

Probability and statistics are the branches of mathematics concerned with the laws governing random events, including collection, analyses, interpretation, and display of numerical data. Probability is distinguished from statistics. While statistics deals with data and inferences from it, probability deals with the random processes which lie behind data or outcomes. There are a few words we connect with probability and they are: chance, randomness, likelihood,  possibility and expectation. 

Probability has its origin in the study of gambling and insurance in the 17th century. Historically the scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. In the early years of the development of mathematics, mathematicians as everybody else believed that God created the world and everything that will happen or will not happen is going to be by God’s will. People didn’t even question this. This is one of the reasons probability didn’t develop straighway with other fields of mathematics. 

 Probability has a dual aspect: on the one hand the likelihood of hypotheses given the evidence for them, and on the other hand the behavior of random processes   such as the throwing of a dice or coins. 

Probable and probability in some modern languages derived from medieval learned Latin probabilis. The form probability is from Old French probabilite (14c.) The mathematical sense of the term is from 1718. In the 18th century, the term chance was also used in the mathematical sense of "probability" (and the probability theory was called Doctrine of Chances). 

The branch of mathematics probability fully developed in the 20th century, building its own methods and theories. 

The modern mathematics of chance is usually dated to a correspondence between the French mathematicians Pierre de Fermat and Blaise Pascal  in 1654. Their inspiration came from a problem about games of chance, proposed by a remarkably philosophical gambler, the chevalier de Méré.  De Méré inquired about the proper division of the stakes when a game of chance is interrupted. Suppose two players, A and B, are playing a three-point game, each having wagered 32 pistoles, and are interrupted after A has two points and B has one. How much should each receive? Games of chance such as this one provided model problems for the theory of chances during its early period, and indeed they remain staples of the textbooks.

Did gambling develop from game playing or did it arise from religious activity? No one knows. We do know that by about 1200 b.c., cubical marked dice  had evolved  from much cruder bones as a  useful device for randomization in games.  Games of chance are probably as old as the human desire to get something for nothing. A game of chance is a game whose outcome is strongly influenced by some randomizing   device, and upon which contestants may choose to wager   money or anything of monetary value. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from a container.

In these  games of chance, maybe some skill elements are playing a final role to win or not, but chance generally plays a greater role in determining its outcome. 

Any game of chance that involves anything of monetary value is gambling.  Gambling is known in nearly all human societies, even though many have passed laws restricting it. Early people used the knucklebones of sheep as dice. Some people develop a psychological addiction to gambling, and will risk even food and shelter to continue.

Today the probability theory is applied in everyday life in risk  assessment and modeling.  The insurance industry in sales and markets use probability theories and some methods to determine pricing and make trading decisions. 

Not everything is as straightforward as the toss of a coin or a die. Many professions rely on probability. We use probability in daily life to make decisions when we don't know for sure what the outcome will be.  Nearly every day we use probability to plan around the weather. Meteorologists can't predict exactly what the weather will be, so they use tools and instruments to determine the likelihood that it will rain, snow or hail. For example, if there's a 60-percent chance of rain, then the weather conditions are such that 60 out of 100 days with similar conditions, it has rained. Meteorologists also examine historical data bases to guesstimate high and low temperatures and probable weather patterns for that day or week.

Athletes and coaches use probability to determine the best sports strategies for games and competitions. 

Probability plays an important role in analyzing insurance policies to determine which plans are best for you or your family and what deductible amounts you need. 

We use probability when we play board, card or video games that involve luck or chance. 

Is the power of probability limited or is it endless? The short answer to "what can probability predict?" is nothing. At least not with certainty. Probability is what we use when we can't predict something. Probability is a good long-term decision-making and forecasting tool we use in the face of uncertainty.

Probability cannot predict the winning lottery numbers. It can't even say if we personally should or should not play the lottery. However, in the face of uncertainty, probability can give you the average value of the lottery ticket, and you can decide for yourself whether or not to buy one accordingly.

Probability can't predict whether it will rain at a specific spot at a certain time. It compares present conditions with past data and says with a certain likelihood whether or not it will rain. Probability gives you a sense of the average and a sense of how much you should expect things to deviate from that average, but it can't tell you what will happen.

Here are some activities for you:

Activity 1: When and on which occasions in your life you make some predictions about some future events? What are your predictions when answering in terms of probability? Can you measure your answer for example with percentage?

Activity 2: Can you name at least five things that will happen this week with great certainty so you measure these events with 1. 

Activity 3: Can you name at least five things that are not possible to happen this week, and measure this with 0. 

In our life we often are wondering whether to do something or not to do. For some reasons we hesitate all the time about  making simple decisions in life. Am I making the right decision? Am I going to regret my decision? People hesitate for different reasons: lack of confidence, lack of courage, overanalysis that stops us from action, tendency to doubt a lot,, feeling overwhelmed, feeling nothing, too many options to choose from and many other reasons that block us. Most of the time we regret if we show lack of action at all. And if it is an instinct for something good, I, for example, always follow it. I get involved in other people’s stories and listen if somebody has pain. I cannot imagine how I would look in the mirror to myself if I refuse to listen when somebody is suffering and I just go away because I am too scared to get involved. I think if somebody is suffering it is our moral duty to listen, this is the least we can do. It is listening that helps most of the time and that does the healing. I never ask myself: What is the probability that I am getting involved in this person’s story? 

Personally, I never regretted I have been impulsive to tell people some nice compliments and some nice words about them to make them feel good. On these occasions I never ask myself: Is it a good idea and what would the other person think of my emotional impulsivity?  I never thought that I needed permission to make people smile and make them feel more confident about themselves. For me it is like an instinct and the probability that I will follow this instink again is certain, it is equal to 1. It requires bravery and care. Sometimes life is much more complicated and there are no precise methods and  even the probability scale is too narrow to offer words and advice on how to act and what will be the outcome for us if we choose one way or another. For example: bullying other people, lack of courage to get involved and to show some moral support, showing indifference when an injustice is happening, examples of racism, examples of censure of freedom of speech and so  many other occasions in which we should not hesitate to show zero tolerance. Everybody should get involved to ensure this is not happening again. It is like making the geography around us more clean in a moral way. If everybody does this, if everybody cleans his circle, can you imagine how shiny planet Earth will be?  Can you imagine how grateful Earth will be? And the amount of magical energy Earth will send back to us, can you imagine? 

It depends on us to make our lives places full of love and kindness! There should never be any doubts and hesitations about it! 

(E. S. Lyubenova; LoveMaths Story for my students)