Posts Tagged ‘Mathematics’

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If I can trust my memory, I first learned about randomness in my 12th standard and it is still one the topics I hate the most. For those people, who are going to attend GATE exams, here is a quick tip from me “better not to attempt questions from probability section; the probability that your answer is wrong is greater than .99“. Doing so will help you to save time for the remaining questions.  Back to the title, this post will be having two sections, the first section is,


Imagine there are 1000 people in a locality, 500 men and 500 women. The probability that a particular man and particular woman will marry (assume that none of them are blood relatives, and everyone SHOULD marry before age of 25!) is less than or equal to 1/500 (that is what I know, I am quite weak in probability). Now, the average sperm-count in the ejaculate of Homo sapiens is about 280 million [1]. And, on an average a healthy women produces about 480 ova in her lifetime. Now, calculate the probability of mating of a sperm among this 280 million (this in fact is much smaller number when compared to the total number of sperms produced by a male during his lifetime, but for the sake of simplicity let us take 280 million) with one of the 480 ova[2]. Even after assuming that, this particular zygote grows into a healthy child, and to a mature Homo sapiens, just see the probability for getting that UNIQUE zygote (or unique individual). There is even randomness during the time when zygote is formed (the crossover, number of crossovers)[3]. That means the probability for you, me and we to live (blog) int his world tends to the value ZERO.

There are two (independent) conclusions for the first section,

(a) You don’t exist at all, because the probability that you were born to earth is ZERO (or near ZERO)

(b) You are really ‘lucky’ to live in this world, hence enjoy to maximum, be productive…….. (not necessarily on bed, I mean to contribute to humankind)

Heads and Tails

Assume that I have tossed an unbiased coin (not the ‘sholay’ coin), the probability to get heads and tails are 50%. Is it just a probability? Is coin tossing always random? I had always thought of coin tossing, just because I don’t want to study probability, even the questions on probability confuse me a lot.

When I toss the coin what happens is, the coin rotates about an axis parallel to its diameter, moves up and comes down because of gravity and makes a number of turns about the axis. The outcome, head or tail, is determined by the initial face and the number of turns it made. The number f turns it made depends on rotational speed of the coin about its axis and the distance through which it moves (let us assume that it is equal to twice the maximum height it reaches in magnitude)

Let us see the various factors (not necessarily independent) affecting the outcome of coin-tossing,

  1. Initial position of the coin – whether it is the tail/head side horizontal to the ground or is it inclined at an angle.
  2. Force with which the coin is tossed – the force I apply to toss the coin determines the rotational speed of coin and the maximum height it reaches.
  3. Point at which I apply the force – or the way you hold the coin while tossing. It determines the speed, the path of motion of coin through air, the rotational speed of coin and the maximum height it reaches.
  4. Mass of the coin – It also have effects on the maximum height it reaches, the rotational speed.
  5. Ambient conditions – Will affect the viscosity and density of air, which in turn affects the rotational speed and maximum attainable height of the coin.
  6. Location (Equator, Polar regions, Earth, Saturn, Jupiter)- The magnitude of gravitational pull or technically, the acceleration due to gravity will definitely have an effect on the above mentioned parameters.
  7. ………. there will be more, I could find only 6.

Hence we can say that, the outcome of coin tossing is deterministic or near-deterministic. Now you can stop saying, “it’s god’s grace that India won the toss today” because you can Mathematically model the coin tossing process and predict the outcome. There will only be a single outcome for same set of conditions. More the number of factors, your model will become more accurate.

In a similar way you can model whole processes that happens in Universe, which are traditionally called random.

[1] http://www2.oakland.edu/biology/lindemann/spermfacts.htm
[2] http://www.drstandley.com/bodysystems_femalerepro.shtml
[3] http://en.wikipedia.org/wiki/Fertilization#Fertilisation_and_genetic_recombination

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