Welcome to a lesson
on the geometric distribution.
There are three main characteristics
of a geometric experiment.
Number one,
there are one or more Bernoulli trials or binomial trials
with all failures except the last one, which is a success.
In other words,
you keep repeating what you're doing
until the first is success.
Then you stop.
For example,
you throw a dart at a bullseye until you hit the bullseye.
The first time you hit the bullseye is a success
so you stop throwing the dart.
It might take six tries until you hit the bullseye.
You can think of the trials as failure, failure, failure,
failure, failure, success, and then you stop.
Number two, in theory,
the number of trials could go on forever.
There must be at least one trial.
And number three,
the probability, p, of a success
and the probability, q, of a failure is the same
for each trial.
P plus q is equal to one,
and q the probability of failure is equal to one minus p,
or one minus their probability of success.
We use the notation shown here
for a geometric probability distribution function.
We read this as X is a random variable
with a geometric distribution.
Where again p is the probability
of a success for each trial.
And one minus p or q is the probability
of a failure for each trial.
Let's look at an example.
Suppose that you are looking for a student at your college
who lives within five miles of you.
You know that 55% of the 25,000 students
do live within five miles of you.
You randomly contact students from the college
until one says he or she lives within five miles of you.
What is the probability that you need
to contact four people?
This is a geometric problem,
because you may have a number
of failures before you have one success you desire.
Also the probability of success stays the same
each time you ask a student
if he or she lives within five miles of you.
There is no definite number of trials,
the number of times you ask a student.
So in this case,
we let the random variable X equals the number
of students you must ask until one says yes.
Next, what values does X take on?
Well, it could take one, two, three, four,
and so on students until one students says yes,
they live within five miles of you.
But doesn't go on forever actually,
because remember 55% of the 25,000 students
do live within five miles of you.
And therefore 45% of the students
don't live within five miles of you.
And 45% of 25,000 is equal to 11,250.
And therefore the maximum number
of students it would take
before one does live within five miles
of you would be 11,250 plus one, or 11,251.
Again, 45% of 25,000 plus one.
This would be,
if you happen to ask every student
that lived further than five miles from you,
that will be 11,250,
and therefore the next one would guarantee
to live within five miles of you.
Next, what are p and q?
P is a probability of success.
Success is when you find a student
that lives within five miles of you,
which is 55%, or is a decimal 0.55.
And q, the probability of failure is equal to one minus p.
In this case one minus 0.55, what is equal to 0.45,
the percent of students
that live further than five miles from you.
Next, what is the probability
that you need to contact four people
before you find a student that says yes,
they do live within five miles of you?
That would be the probability that X is equal to four.
The probability that X is equal to four is equal
to the probability of failure from the first three people
that you contact times the probability
the fourth person is a success,
or does live within five miles of you.
So we would have the probability of failure as 0.45,
times itself three times or raised to the third power,
and then times the probability of success, which is 0.55.
Looking at the formula here in the lower right hand corner,
the probability that X is equal to k is equal
to the quantity one minus p or q raised the power
of k minus one times p.
Notice in our case, k is equal to four.
And therefore the exponent on one minus p or q is three,
and then we have times p.
And now let's go to the calculator
and determine this probability.
Let's round to four decimal places,
which will be approximately 0.0501.
Let's take a look at one more example.
Assume that the probability
of a defective computer component is 0.02.
Components are randomly selected.
Find the probability that the first defect is caused
by the seventh component tested.
How many components do you expect
to test until one is found to be defective?
So in this case, would you have a geometric distribution?
The random variable X is equal to the number
of computer components tested
until the first defect is found.
To find the probability the first defect
is in the seventh component
when you define the probability that X is equal to seven.
Let's start with p the probability of success,
which is the probability
that we find a defective component, which is 0.02,
and then q or one minus p is equal
to the probability of failure,
which is one minus 0.02, which is 0.98.
So the probability that X is equal to seven is equal
to the probability the first six components
are not defective,
which would be 0.98 raised to the power of six,
times the probability the seventh component is defective,
which is 0.02.
Again, looking at our formula,
we have k is equal to seven,
and therefore we have q or the probability
of failure raised to the sixth power,
times the probability of success,
which in this case is that we have a defective component.
And now let's go to the calculator
and get a decimal approximation for this probability.
So we have .98 raised to power of six,
right arrow, times .02
to four decimal places is approximately 0.0177.
And then to answer the question,
how many components do we expect
to test until one is found defective,
we needed to find the mean
where the mean is equal to one divided by p,
which in our case is equal to one divided by 0.02.
0.02 is equal to 200ths or 1/50th.
This is equal to one divided by 1/50th,
which is equal to one times 50 over one or 50.
Which means we expect to test 50 components
before we find a defective component.
Which is why the probability
that X is equal to seven is so low.
The content from this video is
from the "Introductory Statistics Textbook" from OpenStax,
which can be downloaded using the link shown here.
I hope you found this helpful.