March 20, 2018 Tuesday
Bedtime Story
Probability Theory and 𝑒 - 2
Probability Theory and 𝑒 - 2
Jacob Bernoulli had asked the following question to himself: What would be probability for a gambler to lose if he tries his hands on a slot machine that is programmed to pay out once in n times and the gambler plays it n times?
Well, to most average apes the intuitive
answer would be that playing it n times would result in him losing n-1 times
and winning 1 time.
The answer is not simple as that.
Bernoulli came to a remarkable finding
using binomial theorem that if the gambler plays the slot machine n times, then
his probability of winning 0 times after n tries is very close to the limit for
1/e.
1/e = lim n→
This means that were such a gambler to
decide to try out 30 attempts on trying out his lick to win in such a slot
machine, then the probability of his or her losing every bet is 1/2.71828.
This is the probability of failure in his
attempt at winning a hand in slot machine.
This work of Bernoulli spawned a whole new
field of mathematical study in probability and statistics that goes by the name
of Bernoulli trial.
I shall very briefly walk over this subject
as it is related to this great mathematician and once having done that, we
shall have a look at the Bernoulli numbers which was the prime reason for us to
digress into the life and work of this great mathematician.
Any act or experiment or game that can have
only two possible outcomes would be default fall under the category of
Bernoulli trial.
The most classical case of a Bernoulli
trial is flipping of a coin as it can either have a head or tail when it falls
flat on your palm.
Similarly, any live delivery in a maternity
hospital can either be a boy or a girl and hence it would be a fair game for
Bernoulli trial.
In general, any question that has as its
answer as a “yes” or a “no” becomes a Bernoulli trial if it is reiterated.
Independent repeated trials of any
procedure that has exactly two possible outcomes are called Bernoulli trials.
Mathematicians who formally carry out
Bernoulli trials tend to label one outcome as “success” and the other outcome
as “failure”.
The probability of success in Bernoulli
trials is denoted by p and the probability of failure is denoted by q.
Then the sum of the probability of success
and probability of failure is 1 and that is denoted mathematically as: p + q =
1
Moreover, this idea further brings in forth
two more terms in the arena of logic and probability:
1. Mutual Exclusivity
2. Jointly or Collectively Exhaustive
Events
We shall consider this interesting problem
n the nights to come.
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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