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Now, let us visit some basic definitions about probability,
as it relates to the most commonly used concepts in statistics.
Essentially, probability is a measure between zero and
one for the likelihood that something or some event might occur.
For instance, you may hear that the stock markets, the chance of stock market's
rising above some point, or falling below some point is x%,
or you may hear that the chance for rain is 45% tonight.
These are all coming from this very concept of probability.
Essentially, probability is a measure between zero and one, so
45% would be 0.45, the discussion about probability is not complete
without a discussion about random variables.
Essentially, random variable is a quantity whose possible values
depend in some clearly defined way on a set of some random events.
It's a function that maps out outcomes, that is, points in a probability space.
So, probability space essentially is all possible outcomes, If you roll a die,
it can have one out of six outcomes, so that's the probability space there.
If you roll two dice, you can have one out of 36 outcomes where each outcome could
be considered a random outcome.
And probability distribution is a theoretical model that depicts
the possible values any random variable may assume along with
the probability of its occurrence.
We'll define this more with examples using two dice.
So, consider two dice, a die has six faces, and if you roll two dice,
it can assume one out of 36 discrete outcomes.
So, if you were to roll two dice, the probability that
both die will one as the outcome will be 1 + 1 = 2, and
there's only one possibility of getting that and that's one out of 36.
So, here we have two die, one black and one white, and if you were to look at
the possibility of getting one on black and two on white and that's one outcome.
So, that's 1 + 2 is 3 or you can have two on black and one on white,
that's 2 + 1 = 3 again.
So, there are two ways of getting three by rolling two dice, so the outcome or
the probability is two out of 36 possible outcomes that are mapped out here.
So, if you think about the sum of two dice being two,
there's only one possibility out of 36.
Getting a three, you have two possibilities, getting a four,
you have three possibilities, getting a five, you have four possibilities and
so on so forth.
The maximum most frequently possible number to have as a sum of two
dice is seven, and the probability is six out of 36, which is 0.167.
And if you were to sum these probabilities up and
that is 0.02 + 0.056, you get 0.083.
So, if you sum up all these, they all sum up to one and
the probability of getting six or greater than six is 0.58 or
getting less than equal to six is 0.417.
You sum these up, it's 1.028 + 0.972 is 1,
this plus this is 1, this plus this is 1 and obviously 1 + 0 is 1.
The probability of getting 12, that is,
both die show six is 1 out of 36 possible outcomes which is 0.028.
So, the probability sums up to 1 and the probability of getting more than 12
is obviously 0 because the two dice can maximum produce this number.
So, nice way of looking at the way a probability distribution
space is created by rolling two dice.
And if I were to look at the probability of say, getting some number or
less than some number, that's called the cumulative distribution function.
If you were to simply chart this probability
outcomes in a chart, you get this graph here.
So, here we have age as our variable and I created a histogram of age, and
then using the mean value of 48.37 which is the mean average age and
a standard deviation of 9.8 years.
I can fit a theoretical normal distribution with these two parameters.
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