Probability

Here are the 954 words that make upthe section of my dissertation titled "Probability".

I'm not 100% satisfied with it, but it'll have to do for now. There are too many tasks that need to be done.


“Probability is humanity's attempt to understand the uncertainty of the universe,to define the undefinable. A probability is a quantitative measure of the likelihood of a given event." - Amir D. Aczel

In this section, I will talk about how probability helps us to understand the concept of randomness.

Sometimes, when I think of a random event, I imagine something that is totally impossible to predict or understand. Probability shows that some random events are not as unpredictable as people assume. In July, I attended a maths taster day at Queen Mary University of London, called “Inevitable patterns in mathematics”. There were about 50 of us at the taster day. The lecturers, Ben Green and Imre Leader,called out all the months of the year and asked us to raise our hand when he said the month that our birthday was in. The most popular month was September, so they made all the students who had birthdays in September stand up. This included me. They then asked us what date in September we were born in, and if no one else shared that birthday, the person would sit down. When they came to me, I stated that my birthday was on the 17th of September and funnily enough, another girl who was standing up said that she was also born on that day.
I was very surprised that out of a group of 50 people, it was possible that two people shared the same birthday. As Imre Leader later explained, it was almost certain that two people would share the same birthday: The probability that there is a repeated birthday in a group of 50 people is 1-(365/365)(364/365)(363/365)(362/365)...(316/365) which is around 0.97.
Amir Aczel also talk about a similar situation in the book, Chance- the probability that 2 balls land in the same box, if a number of balls are dropped into a number of boxes at random. There are 2 equations that are stated which I found quite interesting:

"1.2 times the square root of the number of categories gives the number of 'balls' required for even odds that at least two share some characteristic.
And:
1.6 times the square root of the number of categories gives the number of 'balls' required for ninety-five percent probability that at least two share some characteristic."

So, if we go back to the Birthday Problem that I came across at the maths taster day, I can now work estimate:
1) 1.2 times the square root of 365 = 22.926, so 23 people have a 50 percent probability of at least one shared birthday.

2) 1.6 times the square root of 365 = 30.57, so 31 people have a 95 percent probabily of at least one shared birthday.

Now it is easy to see why, in my group of 50 people, there was an extremely high probability (97 percent) that we would find a shared birthday.

Now, let us take the classic example of a rolling of a die. Although one can not predict the next number to show up on the die, we know that after 60 rolls, each number should show up roughly ten times. This is because each number on the die has a 1 in 6 probability of being shown. Obviously, this doesn’t mean that every number will actually appear 1 in 6 times. There is a chance that if a die was rolled 60 times, the number 1 would appear every single time, but the probability of that happening is very small, so we know that this event is extremely unlikely. To conclude, it is possible to roughly predict the amount of times a number should appear on a die if the die has been rolled many times. However, it is impossible to predict what the next outcome on the die will be. Or is it?

I asked some physics and maths graduates: “Do you believe with enough research, one will be able to predict a random event (e.g. the outcome of a dice roll) in the future?” Laura Wherity, a maths graduate, answered: “Rolling a die may appear to be random, but in fact it depends on your starting conditions. For example, if you could control the experiment such that the die is always rolled from the same height, at the same angle with the same forces etc. then it should be possible to achieve the same outcome each time. What would appear to be random actually depends on the starting state. Extending this idea, it may be possible to control the starting conditions of other events aswell, so in this sense events that appear 'random' at present may become more predictable in the future as we understand the conditions in more detail.”

Jonathan Wright, another mathematics graduate, said “If we roll a dice in exactly the same way 100 times, 100 times it would give us the same result. If we model the roll of the die, given the starting conditions we could predict the outcome every time.”

So technically, rolling a die isn’t a random event. It is our lack of knowledge of the mechanics of the die which forces us to call it random, although it actually isn’t. Theoretically, it may be possible to one day understand all the conditions of a random event such as a dice roll, and therefore predict the outcome. This may be able to be applied to other such events, and therefore it will be proven that randomness does not exist.


"Probability is humanity's attempt to use pure mathematics to understand the un-understandable. It is our way of trying to learn something about the workings of chance. Chance remains forever untamable, for fate does what it wants to us and to the world around us."

I feel as though I had something else to say, but I can't remember it...

Anyway, TASKS FOR THE WEEKEND/UPCOMING WEEK:
Read Does God Play Dice
Finish writing Random Number Generators and/or The History of Randomness
Email supervisor tomorrow about next meeting.