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Now, let us visit some basic
definitions about probability,

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as it relates to the most commonly
used concepts in statistics.

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Essentially, probability is
a measure between zero and

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one for the likelihood that something or
some event might occur.

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For instance, you may hear that the stock
markets, the chance of stock market's

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rising above some point, or
falling below some point is x%,

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or you may hear that the chance for
rain is 45% tonight.

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These are all coming from this
very concept of probability.

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Essentially, probability is
a measure between zero and one, so

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45% would be 0.45, the discussion
about probability is not complete

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without a discussion
about random variables.

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Essentially, random variable is
a quantity whose possible values

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depend in some clearly defined way
on a set of some random events.

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It's a function that maps out outcomes,
that is, points in a probability space.

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So, probability space essentially is all
possible outcomes, If you roll a die,

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it can have one out of six outcomes,
so that's the probability space there.

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If you roll two dice, you can have one out
of 36 outcomes where each outcome could

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be considered a random outcome.

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And probability distribution is
a theoretical model that depicts

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the possible values any random
variable may assume along with

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the probability of its occurrence.

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We'll define this more with
examples using two dice.

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So, consider two dice, a die has six
faces, and if you roll two dice,

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it can assume one out of
36 discrete outcomes.

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So, if you were to roll two dice,
the probability that

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both die will one as the outcome
will be 1 + 1 = 2, and

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there's only one possibility of
getting that and that's one out of 36.

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So, here we have two die, one black and
one white, and if you were to look at

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the possibility of getting one on black
and two on white and that's one outcome.

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So, that's 1 + 2 is 3 or you can
have two on black and one on white,

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that's 2 + 1 = 3 again.

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So, there are two ways of getting three
by rolling two dice, so the outcome or

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the probability is two out of 36 possible
outcomes that are mapped out here.

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So, if you think about the sum
of two dice being two,

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there's only one possibility out of 36.

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Getting a three, you have two
possibilities, getting a four,

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you have three possibilities, getting
a five, you have four possibilities and

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so on so forth.

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The maximum most frequently possible
number to have as a sum of two

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dice is seven, and the probability
is six out of 36, which is 0.167.

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And if you were to sum
these probabilities up and

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that is 0.02 + 0.056, you get 0.083.

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So, if you sum up all these,
they all sum up to one and

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the probability of getting six or
greater than six is 0.58 or

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getting less than equal to six is 0.417.

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You sum these up, it's 1.028 + 0.972 is 1,

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this plus this is 1, this plus this
is 1 and obviously 1 + 0 is 1.

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The probability of getting 12, that is,

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both die show six is 1 out of 36
possible outcomes which is 0.028.

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So, the probability sums up to 1 and
the probability of getting more than 12

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is obviously 0 because the two dice
can maximum produce this number.

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So, nice way of looking at the way
a probability distribution

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space is created by rolling two dice.

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And if I were to look at the probability
of say, getting some number or

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less than some number, that's called
the cumulative distribution function.

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If you were to simply
chart this probability

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outcomes in a chart,
you get this graph here.

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So, here we have age as our variable and
I created a histogram of age, and

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then using the mean value of 48.37
which is the mean average age and

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a standard deviation of 9.8 years.

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I can fit a theoretical normal
distribution with these two parameters.

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