PROBABILITY

People are always interested in their future, and what will happen tomorrow is sometimes more important to us than what is happening today. Tomorrow is like compensation and a gentle form of revenge for us. Most people are fascinated by tomorrow. What will our life look like? Am I going to be richer and prettier? It is like an escape from today, because today is always guilty of something.

That is why we often try to predict the future. Sometimes our predictions are based on what we know, and we have several possibilities to choose from. In this case, we simply use techniques and methods to determine which possible events are more likely than others. This kind of predictability is the subject of mathematics.

When our predictions are not based on real facts, it is difficult to decide which event has a greater chance of occurring in the future. In this case, it is like playing a blind game with no certain rules.

In this short text, it will not be possible to describe everything that is somehow related to probability. We will simply try to open the door and allow curiosity to come in.

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

Probability is a branch of mathematics concerned with how likely something is to happen. The simplest definition of probability is that it is a number between 0 and 1, where 0 indicates that something is impossible and 1 indicates that something is certain to happen.

A simple example that illustrates this idea is tossing a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are equally probable. The probability of getting heads equals the probability of getting tails, and since no other outcomes are possible, the probability of either heads or tails is 1/2 (which can also be written as 0.5 or 50%).

When dealing with experiments that are random and well defined in a purely theoretical setting (such as tossing a fair coin), probabilities can be described numerically by dividing the number of desired outcomes by the total number of possible outcomes.

For example, tossing a fair coin twice yields the outcomes "head-head", "head-tail", "tail-head" and "tail-tail". The probability of getting "head-head" is one out of four possible outcomes, or, in numerical terms, 1/4, 0.25 or 25%.

Many events cannot be predicted with complete certainty. The best we can do is estimate how likely they are to happen by using the idea of probability.

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5 and 6. The probability of any one of them is 1/6.

Probability is only a guide. It does not tell us exactly what will happen. For example, if we toss a coin 100 times, how many heads will come up? Probability tells us that heads have a 1/2 chance, so we expect about 50 heads. But when we actually perform the experiment, we might get 42, or 52, or another number altogether. In most cases, however, the result will be close to 50.

Some words have a special meaning in probability.

An experiment is a repeatable procedure with a set of possible results. For 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}.

An outcome is a possible result of an experiment. For example, getting a "6".

The sample space consists of all the possible outcomes of an experiment. For example, when choosing a card from a deck, there are 52 cards in the deck. Therefore, the sample space consists of all 52 possible cards.

The sample space is made up of sample points. A sample point is simply one of the possible outcomes.

An event consists of one or more outcomes of an experiment.

Probability and statistics are branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation and presentation of numerical data.

Probability is distinguished from statistics. While statistics deals with data and the inferences drawn from it, probability deals with the random processes that lie behind data and outcomes.

There are a few words that we commonly associate with probability: chance, randomness, likelihood, possibility and expectation.

Probability has its origins in the study of gambling and insurance in the seventeenth century. Historically, the scientific study of probability is a relatively modern development in mathematics. Gambling shows that people have long been interested in quantifying the ideas of probability, but exact mathematical descriptions appeared much later.

There are several reasons for the slow development of the mathematics of probability. During the early development of mathematics, mathematicians, like everyone else, believed that God had created the world and that everything that happened or did not happen was determined by God's will. People did not even question this. This is one of the reasons why probability did not develop alongside the other fields of mathematics.

Probability has a dual aspect. On the one hand, it concerns the likelihood of hypotheses given the available evidence. On the other hand, it concerns the behaviour of random processes such as throwing dice or tossing coins.

The words probable and probability in some modern languages are derived from Medieval Latin probabilis. The form probability comes from the Old French probabilite (14th century). The mathematical meaning of the term dates from 1718. During the eighteenth century, the word chance was also used in the mathematical sense of "probability", and probability theory was often called the Doctrine of Chances.

The branch of mathematics known as probability developed fully during the twentieth century, building its own methods and theories.

Modern probability theory is usually dated to the correspondence between the French mathematicians Pierre de Fermat and Blaise Pascal in 1654. Their inspiration came from a problem concerning games of chance, proposed by the remarkably philosophical gambler, the Chevalier de Méré.

De Méré asked how the stakes should be divided when a game of chance is interrupted. Suppose two players, A and B, are playing a game in which the winner is the first to score three points. Each player has wagered 32 pistoles, and the game is interrupted when A has two points and B has one. How much should each player receive?

Games of chance such as this provided model problems for probability theory during its early development, and they remain classic examples in textbooks today.

Did gambling develop from games, or did it arise from religious activity? No one knows. We do know that by about 1200 BC, cubical marked dice had evolved from much cruder bones into a useful device for randomisation 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 a randomising device and on which contestants may choose to wager money or anything else of monetary value. Common devices include dice, spinning tops, playing cards, roulette wheels and numbered balls drawn from a container.

In these games of chance, some elements of skill may influence the final result, but chance generally plays the greater role in determining the outcome.

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

Today, probability theory is applied in everyday life to risk assessment and modelling. The insurance industry, as well as financial markets, use probability theory and statistical methods to determine pricing and make trading decisions.

Not everything is as straightforward as the toss of a coin or the roll of a die. Many professions rely on probability. We use probability in our daily lives to make decisions when we do not know for certain what the outcome will be.

Nearly every day we use probability to plan around the weather. Meteorologists cannot 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 is a 60 per cent chance of rain, it means that under similar weather conditions, it rained on about 60 out of every 100 comparable days.

Meteorologists also examine historical databases to estimate high and low temperatures and the most likely weather patterns for a particular day or week.

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

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

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

Is the power of probability limited, or is it endless?

The short answer to the question, "What can probability predict?" is: nothing—at least not with certainty.

Probability is what we use when we cannot predict something with certainty. It is a valuable long-term tool for decision-making and forecasting in situations of uncertainty.

Probability cannot predict the winning lottery numbers. It cannot even tell us whether we personally should or should not play the lottery. However, in the face of uncertainty, probability can calculate the expected value of a lottery ticket, and we can decide for ourselves whether it is worth buying one.

Probability cannot predict whether it will rain at a specific place and time. Instead, it compares present conditions with historical data and estimates how likely rain is under those circumstances.

Probability gives us a sense of what is typical and of how much results are likely to vary from that average, but it cannot tell us exactly what will happen.

Here are some activities for you:

Activity 1: Think about the occasions in your life when you make predictions about future events. What are your predictions in terms of probability? Can you express your answers as percentages?

Activity 2: Can you name at least five things that will happen this week with great certainty? Represent these events with the probability value of 1.

Activity 3: Can you name at least five things that are impossible to happen this week? Represent these events with the probability value of 0.

In our lives, we often wonder whether to do something or not. For various reasons, we hesitate when making even simple decisions. Am I making the right decision? Am I going to regret it?

People hesitate for many different reasons: lack of confidence, lack of courage, overanalysing situations until action becomes impossible, a tendency to doubt themselves, feeling overwhelmed, feeling emotionally empty, having too many options to choose from, and many other reasons that hold us back.

Most of the time, however, we regret not acting at all. And when my instinct tells me that something is good, I always follow it.

I have never regretted being impulsive when telling people sincere compliments and kind words that make them feel good. On such occasions, I never ask myself, "Is this a good idea?" or "What will the other person think of my emotional impulsiveness?"

I have never felt that I needed permission to make people smile or to help them feel more confident about themselves. For me, it is an instinct, and the probability that I will follow this instinct again is certain—it is equal to 1.

It requires courage and compassion.

Sometimes life is much more complicated, and there are no precise methods. Even the probability scale is too limited to tell us how we should act or what the outcome of our choices will be.

Think about bullying, the lack of courage to get involved and offer moral support, showing indifference when an injustice is taking place, racism, censorship of freedom of speech, and so many other situations in which we should show zero tolerance.

Everyone should get involved to help ensure that these things do not happen again.

It is like making the geography around us morally cleaner. If everybody did this, if everybody cleaned their own circle, can you imagine how brightly planet Earth would shine? Can you imagine how grateful the Earth would be? And can you imagine the amount of magical energy the Earth would send back to us?

It depends on us to make our lives places filled with love and kindness. There should never be any doubt or hesitation about that.

(E. S. Lyubenova, LoveMaths Story for My Students)