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ProbabilityIntroduction

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Probabilities and likelihoods are everywhere around us, from weather forecasting to games, insurance or election polls. However, in the history of mathematics, probability is actually a very recent idea. While geometry and algebra were studied by ancient Greek mathematicians more than 2500 years ago, the concepts of probability only emerged in the 17th and 18th century.

According to legend, two of the greatest mathematicians, Blaise Pascal and Pierre de Fermat, would regularly meet up in a small cafe in Paris.

To distract from the difficult mathematical theories they were discussing, they often played a simple game: they repeatedly tossed a coin – every heads was a point for Pascal and every tails was a point for Fermat. Whoever had fewer points after three coin tosses had to pay the bill.

One day, however, they get interrupted after the first coin toss and Fermat has to leave urgently. Later, they wonder who should pay the bill, or if there is a fair way to split it. The first coin landed heads (a point for Pascal), so maybe Fermat should pay everything. However, there is a small chance that Fermat could have still won if the two next tossesnext coin toss would have been tails.

Pascal and Fermat decided to write down all possible ways the game could have continued:

HHH

Pascal wins

HHT

Pascal wins

HTH

Pascal wins

HTT

Fermat wins

All four possible outcomes are equally likely, and Pascal wins in threefourtwo of them. Thus they decided that Fermat should pay 3/4 of the bill and Pascal should pay 1/4.

Pascal and Fermat had discovered the first important equation of probability: if an experiment has multiple possible outcomes which are all equally likely, then

Probability of an event = Number of ways the event could happenTotal number of possible outcomes.

In our example, the probability of Pascal winning the game is 34=0.75, and the probability of Fermat winning the game is 14=0.25.

What are Probabilities

A probability is a number between 0 and 1 which describes the likelihood of a certain event. A probability of 0 means that something is impossible; a probability of 1 means that something is certain.

For example, it is impossiblecertain that you will meet a real life dragon, and it is certainimpossible that the sun will rise tomorrow. The probability of a coin landing heads is exactly in the middlethe same.

The probability of rolling a 6 on a die, or picking a particular suit from a deck of cards is lessmore than 0.5 – which means unlikely. The probability of a good football team winning a match, or of a train arriving on time is moreless than 0.5 – which means likely.

dragondicecardscoinsfootballtrainsun

Here are some more events: drag them into the correct order, from likely to unlikely:

You throw a die game_die and it lands on 6.
Penguins penguin live on the North Pole.
It’s going to rain rain_cloud in November.
A baby will be born in China today. baby_bottle
You buy a lottery ticket and win the Jackpot tada.
A newborn baby will be a girl girl.

We often use probabilities and likelihoods in everyday life, usually without thinking about it. What is the chance of rain tomorrow? How likely is it that I will miss the bus? What is the probability I will win this game?

Tossing a (fair) coin has two possible outcomes, heads and tails, which are both equally likely. According to the equation above, the probability of a coin landing heads must be 12 = 0.5, or 50%.

Note that this probability is in between 0 and 1, even though only one of the outcomes can actually happen. But probabilities have very little to do with actual results: if we toss a coin many times we know that approximately halfexactly halfallnone of the results are heads – but we have no way of predicting exactly which tosses landed heads.

Even events with tiny probabilities (like winning the lottery tada) can still happen – and they do happen all the time (but to a very small proportion of the people who participate).

Probabilities also depend on how much each of us knows about the event. For example, you might estimate that the chance of rain today is about 70%, while a meteorologist with detailed weather data might say the chance of rain is 64.2%.

Or suppose that I toss a coin and cover it up with my hands – the probability of tails is 50%. Now I peek at the result, but don’t tell you. I know for certain what has happened, but for you the probability is still 50%100%not 50%.

There are many different ways to think about probabilities, but in practice they often give the same results:

classical probability

The classical probability of landing heads is the proportion of possible outcomes that are heads.

frequentist probability

The frequentist probability is the proportion of heads we get if we toss the coin many times.

subjectivist probability

The subjectivist probability tells us how strongly we believe that the coin will land heads.

Remember that while probabilities are great for estimating and forecasting, we can never tell what actually will happen.

Predicting the Future

If we roll a dice, the result is a number between 1 and 6, and all outcomes are equally likely. If we roll two dice at once and add up their scores we can get results from up to . However, in this case they are not all equally likely.

Some results can only happen one way (to get 12 you have to roll + ) while others can happen in multiple different ways (to get 5 you could roll + or + ).

This table shows all possible outcomes:

2
3
4
5
6
7
8
9
10
11
12

The most likely result when rolling two dice is 7. There are outcomes where the sum is 7, and outcomes in total, so the probability of getting a 7 is 636=0.1666.

The least likely outcomes are 2 and 12, each with a probability of 136=0.0277.

It is impossible to forecast the outcome of a single coin toss or die roll. However, using probability we can very accurately predict the outcome of many dice.

If we throw a die 30 times, we know that we would get around 16×30=5 sixes. If we roll it 300 times, there will be around 16×300=50 sixes. These predictions get more and more accurate as we repeat the predictions more and more often.

In this animation you can roll many “virtual” dice at once and see how the results compare to the predicted probabilities:

Rolling Dice

${ probTable(d) }

We roll ${d} dice at once and record the SUM of their scores. The green lines represent the probabilities of every possible outcome predicted by probability theory and the blue bars show how often each outcome happened in this computer generated experiment.

Notice how, as we roll more and more dice, the observed frequencies become closer and closer to the frequencies we predicted using probability theory. This principle applies to all probability experiments and is called the Law of large numbers.

Similarly, as we increase the number of dice rolled at once, you can also see that the probabilities change from a straight line (one die) to a triangle (two dice) and then to a “bell-shaped” curve. This is known as the Central Limit Theorem, and the bell-shaped curve is called the Normal Distribution.