Power of Geometry – Using area in continuous probability

We encounter probability everywhere in our life. From estimating the chances of our teachers catching us dozing off in class to the probability of us meeting Ariana Grande on the streets, probability is inherent in our daily lives.

The inspiration of this introduction of using geometry in probability came to me just this afternoon. I was walking towards the bus stop when 2 buses (that share the same route) sped past me, back to back. (It was a horrible looking at them cruise past me just like that, but that’s another story all together.) 2 girls in front of me also noticed this and one of them asked the other a question which I thought was interesting. She asked what the probability (of the 2 buses going to the same bus stop back to back happening) was. After which, she attempted to suppose the condition that if 3 buses were to be employed within some interval of an hour, to tackle the question of finding the probability that 2 of them will come consecutively to the bus stop. At that moment, I was literally shouting in my head, “Use Areas to find out!” while calculating the chances simultaneously. So, here’s the story behind my “Comeback” post and I hope you will enjoy it. 🙂

The best way to start any form of introduction is to start trying your hand on the problem itself! How would you find the probability of that event occurring???

First, we identify some variables we know and try to quantify others. We know we have 2 buses, let them be A and B respectively. We also know that we are concerned with the chance of 2 buses being within a certain range of distance from each other in an interval of an hour (The key here is the interval of 1 hour). We need to quantify what’s considered as “back to back” or what the “certain range of distance” mean. So suppose I define it as 2 buses being 20 minutes or less apart (meaning that, assuming the buses move at equal speed, it takes 20 or less minutes for the bus lagging behind to catch up to the position of where the bus leading was). Why 20, well for easy calculations and drawing as you will see.

Now that we’ve gotten our definitions and variables ready, let’s try to use some techniques we learnt in school. Hmm… But I’m stuck here… The buses can come at any time within the 1 hour interval constraint; it can come at 0.5h or 0.488888h or \frac{1}{\pi} h and so on. The number of timings it can come is infinite! As such, we say the probability in this case is not discrete; instead it is on a continuous spectrum.

What’s a better model than using areas and geometry?

We see that a line is similarly made up of infinite points so that will be useful. So we can represent the time a bus arrives as a number line. So for 2 buses we will have 2 lines. We want to find the chances that 2 buses come consecutively within 20 minutes (or \frac{1}{3} h of distance apart.

Blog post 1

We can see this problem as finding the proportion of the number of points (or rather proportion of area set of these of points since the no. are infinite) that fall within distance \frac{1}{3} from each other. This is depicted as the blue area in the Diagram. Thus, the probability of bus arriving consecutively within the 20 minutes range is Area of Blue Region/Total Area. 

From here, the total area of box is simply 1 \times 1 = 1 and area of blue region = 1 - \text{Area of 2 white triangular regions} = 1 - 2(\frac{1}{2}) (\frac{2}{3})^2 = \frac{5}{9}. Thus the probability of 2 buses being at most 20 minutes apart from each other is \displaystyle \frac{\frac{5}{9}}{1} = \frac{5}{9}.

Can you think of a generalisation using the Area method for a situation where the “certain range of distance” is defined as k minutes apart for an interval of n hours for some real n, k.


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