Posts Tagged ‘mathematics’

Math is probably for you

November 22, 2009

Math can be really fun. Seriously.

This post is the 2nd in a series of posts I’m planning to have about why math is such a beautiful, useful, and awe-inspiring subject, and that a lot of us can do math (advanced/seemingly difficult math even). Math is such an integral part of humanity since our cave dwelling days, and much more so now in most of our technology driven lives. Previously I wrote about how even advanced math, particularly advanced geometry, can be easily tackled with just your imagination. This time it’s about probability. I can just imagine some of you cringe at the thought of math, let alone probability. But I’ll try to show you that often times, logical reasoning is all that it takes to wrap your head around probabilities, even the ones that confound a lot of brilliant people, even some mathematicians themselves. In fact, we’ll end this article with a simulation of a game/game show. Not bad huh? 🙂

Probability and people

In a nutshell, probability is the area of math which deals with the likelihood of an event happening. It is usually expressed as a number, whether a fraction or a decimal, between 0 and 1, with a probability of 1 meaning the event will surely happen and a probability of 0 meaning the event won’t happen.

Now, don’t be too hard on yourself thinking that probability is too hard for you, unlike most of the human population. In fact, probability is one really confounding area of math and problems in it that seem to be easy in hindsight, turn out to be deceptively difficult or tricky, even for  mathematicians, teachers, and other brilliant men and women around the globe. In fact a lot of us have trouble wrapping our heads around probabilities. You mix that with human hopefulness and also the difficulty of grasping very large numbers and what you get is the staggering number of people around the world falling in line to get their lotto tickets so they could win the multi-million prize money.

In fact, if we do the math, in a typical 6/49 game of lotto (6 unique numbers chosen out of 49 numbers, where the order of the 6 numbers is not important) we find that your chances of winning today after buying that lotto ticket is 1 in about 14,000,000. So if Lucy (one of the earliest hominids/proto-humans known to us) or her people, or perhaps even Neanderthals started betting on the lottery at the beginning of their lives, some of them should be millionaires by now. That’s how bad we are at assessing odds, especially coupled with large numbers. So when you go buy that lotto ticket later, I’m afraid the odds are so much against you.

However, I’ll discuss next a particularly perplexing probability problem pondered by people, even brilliant ones, and found the solution to be deceptively trivial after all. Actually, even after you get the explanation, from a practical standpoint it doesn’t seem like so. But the logical reasoning will quite surely buy you out. But don’t fret, all you need again is imagination and logical reasoning. 🙂

Game time

Some of you may have heard/read about the American game show Let’s Make a Deal. The Monty Hall problem (MHP) was named after the show’s host. Simply stated, the rules of the game are as follows:

The game master (GM), has 3 doors: 2 with goats behind them and one with a car behind it. The GM lets you choose one door, which you think holds the prize car behind it. Since the GM’s job is to make you and the audience excited and enjoy the game, the GM opens another door. But since the GM knows the placement of the goats and the car i.e. which door has which item behind it, the GM opens a door which has a goat behind it. Now, the GM poses a question to you: Do you or do you not want to change the door you initially picked i.e. the GM gives you an opportunity to stay with the door you originally picked, or to choose the other door, knowing that one of the doors, which the GM opened, has a goat behind it.

The GM in the show is of course Monty Hall (MH). Now, you’d most probably think that since there are only 2 doors left unopened, that the probability of getting either a goat or a car is now 50/50 or 50% right i.e. it doesn’t matter whether you switch doors or not?


In fact, however counterintuitive this may seem, your chances of getting the car at this point of the game doubles if you decide to change the door you initially picked. How? Let’s find out shall we? 🙂

Goat, Car, Goat

Now let’s strap on our imagination and logical reasoning caps to find out how the probability of getting the car increases two-fold if you switch your chosen door, and that it’s not a 50/50 chance of getting the car once a door with a goat has been opened by the GM.

Monty Hall problem

Monty Hall problem

One way of looking at how this counterintuitive probability problem is correctly tackled is by taking the possibility of the events one at a time (refer to the figure above please). In this scenario we show that when you switch doors, you always double your chances of winning. Here’s how:

1. First event, say you picked a door and it happened to have the prize car behind it.  Regardless of which door the GM opens, switching in this case either gives you goat A or goat B i.e. you lose the prize car. Out of the 3 possible scenarios (2 of which are listed right after this one), in this one event/case do you lose the prize car.

2. Second event, you choose a door with a goat (goat A) behind it. The GM opens a door again with a goat (goat B) behind it. If you switch in this case, you get the car. This event, wherein you get the car by switching, is one event which you get the prize car. Score one for you. 🙂

3. Third event, you choose the 3rd door with a goat(goat B) behind it. The GM again opens a door with a goat (this time, goat A). So when you switch, you get the car again. Yay. 🙂 This event, wherein you again get the car by switching, is another event which lets you take home the prize.

So what did we get from all this? We saw that out of 3 events/cases of picking either of the 3 doors, you always get 2 events (event 2. and 3.) which favor switching and which lets you walk away with the prize (or in this case, drive away with the prize). So the odds of getting the car/prize in the MHP is not 50% as a lot of us would initially assume, but instead, is really 2/3 or approximately 66.7%.

It can take a while to sink in, but the reasoning/explanation is quite logical and sound.

Try it out!

I actually tried this out with my mother and at another time with my younger brother. What I did was I got 3 opaque plastic cups (simulating the doors) and 2 toy cows (no goat toys in our house at that time) and 1 robot toy that transforms into a car (not bad for a prize no?). I made them act as a GM at one time, with me being the game contestant. Of course to prove my point I always switched. We did this about 20 times and I got the prize car (or robot) at around 14 times out of the 20 (roughly 2/3 of 20). Then I acted as a GM and they acted as the contestant. Then their job was not to switch doors (or cups), just to prove my point that you get the prize more often than not (2/3 of the time remember?) by switching instead of staying with your original door/cup.

They even asked me if I was doing a magic trick on them. I told them it was the power of mathematics and of logical thinking. 🙂 Imagine what much more primitive, let’s say Bronze-aged men, would think of me, with this knowledge, even without modern devices like a cellphone. Perhaps they’d think of me as an oracle or even a god. 🙂

Great, great. But what’s the use?

I think one important thing we can get from this (other than to show you that you can do maths you thought were too hard or complicated for you) is that with math, we can make decisions in our lives (sports betting, lottery, game shows and so on) with more clarity, logic, and sound reasoning, instead of just blind optimism.

If you didn’t get the logic on how to win the game at first glance, or if you thought it was 50/50, don’t be ashamed, a lot of people (some brilliant even) fell for it too. In fact, out of 228 subjects in a study, only 13% chose to switch, and that the rest (87%) assumed that the switching didn’t matter since the likelihood of getting the car out of the 2 unopened doors are equal (research by Mueser and Granberg, 1999).

Quoting cognitive psychologist Massimo Piattelli-Palmarini

“… no other statistical puzzle comes so close to fooling all the people all the time”


“that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.”

So, not bad eh? Still think math (or at least those areas you think are too advanced or complicated for you) isn’t for the average person? If so, then look forward to my next posts about math. 🙂

References, resources, and further reading

Towards healthier skepticism: Correlation does not imply causation

October 9, 2009

This post will attempt to repeat, clarify, and elucidate the need for the remembrance and understanding of the phrase “correlation does not imply causation”. Scientific studies will be given, and the words in the phrase, which vary in meaning depending on usage, will be defined accordingly.

Scientific studies

Please take a moment to go through the following actual, summarized scientific research results:

1) In a previous scientific research using quantitative assessment, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD.

2) From a study at the University of Pennsylvania Medical Center, young children who sleep with the light on are much more likely to develop myopia in later life.

We will get back to them in a moment. Now we focus on correlation or co-relation, and why scientists, statisticians and skeptics, at the very least, should always maintain and promote the phrase “Correlation does not imply causation”.


Reflections of the implications of the Pigeonhole principle in our lives

February 17, 2008

Readability test scores for this post are as follows:

Flesch Reading Ease: 70.52
Flesch-Kincaid Grade Level: 7.00
Automated Readability Index: 7.00

For more info about readability tests, check out my post about those here.

Math is fun in my opinion. It has, for the longest time, including Science, been my favorite subject.

Anyway, this post is about a portion of mathematics which is more popularly known as the pigeon hole principle. The pigeonhole principle states that, given k+1 or more objects (k being any positive integer) which are to be placed on k number of holes, there is at least one hole which will contain two or more of the objects. Or to generalize the definition in a mathematical way:

if M objects are placed into j holes, M greater than or equal to j, then there is at least one hole containing at least k number of objects, where k is equal to the ceiling( M / j ) .

For example, if I have 12 pigeons to be placed into 11 pigeonholes, there will be at least one pigeonhole which will have 2 pigeons inside. This is so because the ceiling value k, the smallest integer value not less than (M / j) , is 2 since 12 divided by 11 is approximately 1.09 whose ceiling value k is equal to 2

“So what?”

you may say. What would possibly be in the pigeonhole principle that could interest me? Well, a lot actually, same as a lot of other seemingly ‘impractical’ mathematical theories for our every day lives.

1 If we take the upper limit on the average number of hair on the human head (nice alliteration) to be about 150,000, and say a given city has a population size of 500,000, there would be at least 4 people with the same number of hair on their heads. This happened because taking the population size (500,000) as the pigeons and the maximum possible number of hair on the human head (150,000) as the pigeonholes, k is found to be equal to 4. Isn’t that amazing to know? (okay sorry, I know not everyone will find that interesting).

2 Take a pizza party for example: If I were to treat four of my friends so that the five of us will eat a pizza pie of 12 slices, at least one among us will be eating 3 pizzas, while the rest only eat 2. This assumes of course that everyone eats a whole slice of pizza. Now isn’t that worth knowing at times like this? (^)__(^)

3 Another example is when you want to get say, candies from those vending machines which shows you their contents but you can’t pick any flavor in particular (i.e. the machine gives the candy to you at random). Say you can see that there are only two candy flavors left: orange and lemon, and there are 12 of each. You will know (using the pigeonhole principle) that you can get at least 2 candies of the same flavor for sure by buying (not 13 candies) at least 3 candies only! This is because in the event that the first 2 candies you get are of different flavors, you know for certain that the third one will be one of the 2 flavors, giving you at least 2 candies of the same flavor!

However, if you want at least 2 of a specific flavor, say, lemon flavor, it becomes a bit different: you need to get at least 14 candies to be sure that you get 2 lemon flavored candies.

4 Say you have a shop/store with (doesn’t have to be the exact number, since using the principle will still give you the correct answer) 50 aisles, 85 horizontal locations on each aisle, and 5 shelves. The products you sell are stored in bins on horizontal locations on the shelves of each aisle. What is the least number of products you would buy to ensure that at least 2 products are stored in the same bin? That is, based on the above definition, what should be M so that for j = 85 x 50 x 5 = 21250, k = 2 ? Surely enough, M would be equal to 21251 since taking the ceiling of 21251 / 21250 would be equal to 2.

5 If you collect/put together 13 people randomly (or even purposefully), by the pigeonhole principle, there are at least 2 people (pigeons) from that group who were born on the same month, since there are only 12 months (pigeonholes). And, much more obvious, is that among a group of 8 people (randomly/non-randomly chosen), at least two among them were born on the same day.

The important thing to remember here is to figure out which among the objects you’re dealing with is the pigeon and which is the pigeonhole.

There you go! There are tons more of examples on the daily applications of the pigeonhole principle, from computer networks, to train stations, to supermarkets, to art classes, etc. Anybody disagrees with me? (^)__(^)

Btw, comments/suggestions/questions/corrections are welcome, as long as they come in a calm and ruly way.