now let us visit some basic definitions about probability as it relates to the
most commonly used concepts and statistics essentially probability is a
measure between 0 and 1 for the likelihood that something or some event
might occur for instance you may hear that the stock market's the chance of
stock markets rising above some point or falling below some point is X percent or
you may hear that the chance for rain is 45% tonight these are all coming from
its very concept of probability essentially probability is a measure
between 0 and 1 so 45 percent would be 0.45 the discussion about probability is
not complete without a discussion for about random variables essentially a
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 1 out of 6 outcomes so
that's the probability space there if you roll two dice you can have 1 out of
36 outcomes where each column could be considered a random outcome and the
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 phases 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
would have one as the outcome would be one plus one two 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 and white and that's one outcome
and so that's one plus two is three or you can have two on black and one on
white that's two plus one three again so you have two ways of getting three on by
rolling two dice of 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 the three you have two
possibilities getting a four you have three possibilities getting a five you
have a four possibilities and so on so forth the maximum most frequently
possible number to have as a sum of two dice is 7 and the probability 6 out of
36 which is 0.167 and if you were to sum these probabilities up and that is
point zero two plus point zero five six you get point zero eight three so if you
sum up all these they all sum up to 1 and the probability of getting six or
less than greater than six is 0.58 or getting less than equal to six point
0.47 sum these up it's 1 0.028 plus 0.4917 is 1 this Plus
this is 1 this Plus this is 1 and obviously 1 plus 0 is 1 the probability
of getting 12 that is both die shows 6 there is 1 out of 36 possible
outcomes which is .028 so the probability sums up to 1 and the
probability of getting more than 12 is obviously 0 because the two dies 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 and getting some number or less than some number that's
called the cumulative distribution function if you were to simply chart
this probability and 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 point three seven which is the minimum mean every gauge and a
standard deviation of nine point eight years I can fit a theoretical
normal distribution with these two parameters lastly we can compute the
probability of an event or value being achieved so what is the probability of
an instructor getting a teaching evaluation of greater than 4.5 you know
the mean is around 4 so this would be probability the shaded area that you're
looking at and and the way you do it is you take 4.5 which is the value subtract
the mean value which is 4 and then divided by standard deviation which is
0.55 you get the statistics at 0.91 and if you were to take a
standard normal table or just compute the statistics in SPSS you get an 18
point to probable 18.2% probability essentially the probability of getting a
teaching evaluation score of 4.5 and greater than 4.5 is 18.2%