Probability

April 30, 2022

What was the probability that Newton was hit by a coconut?

Remember when you were in school and you were being taught Newton’s laws in Physics? It always started with the story of Newton sitting under an apple tree and how when an apple fell on his head the idea of gravity came into his mind. While studying his theories, oh how we wished he sat under a coconut tree instead!

coconut_treee

Now what would have been the probability of that happening?.Well considering that the UK’s climate is not suitable for coconuts to grow, there was barely any chance of that happening. So yes it was inevitable and we are stuck with Newton’s laws forever now. Well off course it has played a crucial role in shaping our understanding about the workings of nature.

How did it all start?

dice

It all began when a French gambler Chevalier de Méré came across a problem where he wanted to know which of the two games he stood the higher chance of winning at- In Game 1 he had to throw a fair die four times and if he gets 6 he wins. In game 2 he had to throw two fair dice for maximum twenty-four times and if he gets double sixes he wins.

To find the answer Chevalier played for many times only to find that he had a greater chance of winning if he played Game 1.

We will come to the problem again in a later part of the article and see how we can find the answer without carrying out an experiment several times.

What is probability?

If we want to measure weight we measure it in Kilograms, Height in centimetre or inches or metres. But how do we measure probability? Is it even a real quantity? Well it seems that probability is not a real quantity. But we can indirectly measure it using numbers between 0 to 1.

Philosophers and statisticians have some famous suggestions which tell us what those exclusive numbers mean in probability:

Classical Probability: This is equal to the ratio of the number of outcomes favouring the event and the total number of possible outcomes. Here we make the assumption that outcomes are all equally likely. For example rolling dice or tossing coins.

Enumerative probability: Remember the problem where we had a bag and the bag contained 3 red balls and 4 white balls, and we had to calculate the probability of getting a red ball or a white ball? With the idea of random choice from a physical set of objects, we can safely say that it is an extension of classical probability.

Long-run frequency probability: If you conduct a coin tossing experiment for infinite numbers of time the probability of getting head or tail will be 0.5 . It won’t change. Doesn’t matter how many times the experiment is carried out.
This is based on the proportion of times an event occurs in an infinite sequence of identical experiments. Chevalier’s problem also comes in this category.

Subjective or ‘personal’ probability: This is a specific person’s judgement about a specific occasion, based on their current knowledge, and is roughly interpreted in terms of the betting odds that they would find reasonable. That means any numerical probability is essentially constructed according to what is known in the current situation.

Different experts prefer different alternatives to describe probability.

Probability is the result of Randomness. With any random phenomenon, the probability of a particular outcome is the proportion of times that the outcome would occur in a long run of observations.

Time to Know the Rules of Probability.

The probability of an event is a number between 0 and 1.

Complement rule says that probability of an event not happening is given by one subtracted by probability of event happening.

The addition, or the OR rule says to add probabilities of mutually exclusive events to get the total probability.

The multiplication, or the AND rule says to multiply probabilities to get the overall probability of a sequence of independent events occurring.

Now let’s get back to the problem and find why Chevalier has a greater probability of winning Game 1 than Game 2.

In game one we have to throw the dice 4 times and if we get six at least once we win the game.
So let’s answer a few questions: what is the probability of getting six on the dice?
It will be ⅙ using the definition in classical probability. So indeed the probability is between 0 and 1.

Now let’s ask the 2nd question: what is the probability of not getting a six? Using complement rule we get 1 – ⅙ which is ⅚.

Next question will be what is the probability of not getting a six after we throw the dice four times? It will be ⅚ * ⅚ * ⅚ * ⅚ = 0.48

And the final question: What is the probability of getting a six at least one time after we roll the dice four times? Well it is 1 – 0.48 = 0.52.

If we follow the same procedure to find the probability of winning Game 2 then we will get 0.49 as our answer.

So it is obvious that Chevalier had played Game 1 as there was a greater chance that he would have won Game 1 several more times as compared to Game 2.

Conclusion:

application

Though Probability was invented because the gambler had a problem but it has its application in almost every field. Probability has been extensively used in Academics, computation, Stock Market, Businesses etc.
In machine learning we use Probability theories for Regression, Classification, or minimising errors.
People say the stock market is equivalent to gambling which is partially true if you invest without doing any research.
We use probability to avoid or to prepare for any natural disasters. It is used in weather forecasting or to predict the annual growth of a country’s economy.
In short, probability is used to predict the future.

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