March 21, 2018 Wednesday
Bedtime Story
Mutual Exclusivity and Collectively Exhaustive Events
Last night I had left you with two new
terminologies from probability theory and logic: 1. Mutual Exclusivity 2.
Jointly or Collectively Exhaustive Events
Mutual exclusivity is one of the corner
stone principles of logic and probability theory wherein two propositions or
events are regarded as mutually exclusive if they both cannot be true at the
same time.
For instance, when a child is born it can
either be a male or female (barring aside several chromosomal abnormalities).
Being a male child by default excludes it
from becoming a female child.
That makes the sex of a child, that is male
or female mutually exclusive.
Now what do we mean by jointly or
collectively exhaustive events?
Collectively exhaustive events refer to a
collective set of events in the probability theory and logic when of all the
events in the set one has to occur.
In a case of simple toss of a fair coin,
since the outcome has to be either a heads or a tails, the two then become
collectively exhaustive events.
In a case of roll of a dice that is
six-sided, the outcome can be either a 1 or a 2 or a 3 or a 4 or a 5 or a 6 and
thence all these six become collectively exhaustive events.
So how do these terms fit into the concept
of Bernoulli trial?
As discussed earlier, the sum of the probability
of success and probability of failure is 1 or p + q = 1.
Here p or “success” and q or “failure” are
complimentary events and thus they are both mutually exclusive and collectively
exhaustive events.
You would have often heard people talking in
terms of odds in matters concerning chances or probabilities.
The idea of odds also arises from the
concept of Bernoulli trials and the idea of p + q = 1 can be stated in terms of
odds.
Given the probability p of success and q
for failure, the terms that are used in terms of odds are odds for and odds
against.
The odds for essentially is a ratio of p:q
and the odds against means ratio of q:p.
Mathematically, it can be stated as:
of = p/q
oa = q/p
Finally, we now come to the idea of
Bernoulli numbers that too appeared in the book Ars Conjectandi in relation to
the discussion of exponential series.
It is a reflection of the European success
in the arena of world history that today I write of these numbers as Bernoulli
numbers; a Japanese by the name of Seki Takakazu too had discovered these very
numbers much earlier and had even got it published it a year earlier than Jacob.
Stay tuned to the voice of an average story storytelling chimpanzee
or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the International
Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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