November 21, 2018 Wednesday
Bedtime Story
How Bayes Framed the Problem
But these developments (‘Monty Hall
Problem’, the ‘Three Prisoners Problem’ and the Bertrand’s Box Paradox) came much
later and Bayes himself did not see the implication of his work to be so wide.
In his essay Bayes framed his problem in
the following manner:
He considered a series of independent
experiments.
The outcome of these experiments can either
be success or failure.
Let the probability of success be donated
by the small letter ‘p’ and that it has to lie somewhere between 0 and 1.
He also supposed p to be an uncertain
(besides unknown) quantity whose probability of being at any point in the
interval between 0 and 1 is the length of the interval.
By itself the chances of outcomes of a
success or failure in these experiments are equal.
But if large number of observations are
made and we get to know the number of successful outcomes (let it be
represented by k) by observing some variable numbers of trials (let it be
represented by n) then the value of p will change.
The probability of success p will be much
higher than before meaning that the success in the next trial would be more
probable than earlier.
This much he simply stated but did not
question.
The question that he went on to pose was
this:
What is the conditional probability
distribution of p after having known the number of successful outcomes k from n
number of trials seen?
He came with a mathematical solution for
it.
In words it can be stated as follows:
“Given the number of times in which an
unknown event has happened and failed…
The chance that the probability of its
happening in the next single trial lies somewhere between any two degrees of
probability that can be named.”
This might be hard to interpret so Bayes
did it with an example of a man trying to guess the chance of winning a lottery
by observing its lucky draws and failed draws.
Bayes says that if the man has observed 11
draws out of which 10 were blanks and one turned out to be a win, then the
value of p will fall somewhere between 9:1 to 11:1.
But then what happens if the person
interested in winning the lottery continues to make observations.
Suppose after 22 observations he sees that
there have been 20 blanks and 2 wins.
Later he observes 40 blanks and 4 wins and
this goes on for some time until he arrives at the figure of 10,000 blanks and
1000 lucky wins.
This is of course hypothetical but the
point is not that.
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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