BAM! Mr. Tarrou
Let's take a look at a setting that sounds almost like a binomial distribution or a binomial setting
That is a geometric setting so you can read the questions and again for every trial or observation
There's maybe two possibilities the success and [a] failure
But every time you read a problem like shooting free-throws
Answering questions on a test where there's two possibilities of success or failure right or wrong
It is not automatically binomial
And it actually could be neither binomial or this particular setting we're going to talk about which is the geometric [stating]
Any binomial setting there are a set number of trials. I'm going to ask 300 people a particular question. I'm going to take 50 free-throws
I'm going to take this test, and there's 30 questions on it
Whereas in A geometric saying you go until your firm success
I'm going to ask people a question until I get my first person to say yes
We're gonna play the laundry and I want to keep playing some I get my first win
I'm going to have children. I'm going to keep going until I have my first boy or my first girl. That's
[a] geometric setting so geometric setting has four checks, just like binomials there are two outcomes
Just like a binomial setting all the observations must be independent
just like a binomial setting
You must have a constant probability of success, so this is like
[you're] [not] taking a multiple-choice
Question and some problems have four choices some problems have five some problems are true
And false which only give you two chances of success you're not getting help in between your trials every trial has to have the same
Probability of success and the only difference between a geometric setting in the binomial setting is this fourth check up label
Which is the variability of interest is not?
You know like a binomial
Settings is a set number of trials well here the variable of interest is [-] the number of trials it takes
before you get your first success, so
Let we go to free-throws. I'm shooting free-throws on an attempt to shoot until I make my first basket
So I might miss miss miss and then finally make a basket that would be
An example of a geometric setting where I had to go to my fourth trial before I made my first basket
No need for many fancy formulas when you're dealing with the geometric setting because there's only one
Combination of observations you have to fail and the idea is you're going until your first success
So all the initial trials or observations need to be labeled as failures and your last
Observation your last trial is that first success that you actually?
Achieve all right, so let's go to page two
okay, so
Let's finish let's start doing some examples probability my first success is on my end trial
Well, I'm giving you a Ford leaving
oh I just said you don't really need to have one but most students do understand
Probabilities for geometric settings right off the bat the probability that x equals n
The probability your first success is obtained on your end trial is
the probability of basically failure raised to the [N] [minus] 1 power
Rate and then multiplied by P. So basically like the basketball example
I just said my chance of failure, and I miss miss Miss and
That's three and then on my fourth trial. I get my success
So an example even I just gave you one an event has a 15% probability of occurring
what's the probability that your first success occurs on the
Seventh trial and I'm going to keep this on the basis of this example through the rest of these notes
So keep a note of that
Well if your first success is going to occur on your seventh trial then the first six must be labeled as or considered failure
So the probability that x seven the seventh trial observation is the first time or [the] first success you have
That's one minus 0.15
To the seven minus one power
times
0.15 and then it comes out to be point [zero] five seven so there's a seven of five point seven percent probability
You observe your first success on the seventh trial if your probability of success is fifteen percent
let's take that same setting and make a pDf table out of that geometric setting so we have an x
labeled as the number of trials to get your first success and
p of x your probability of that occurring
So you can either work out this formula or you can go to the distribution menu of your calculator [I]?
Think that should be second distribution
pungent menu on your calculator do geometric Pdf
probability success and and again the number of trials until
you get your first success and the probabilities come out to be point one five point one two three and
So on you'll notice that these probabilities are shrinking all the way through this PDf table
And that is because the probability of success [is] point one. Five you're constantly multiplying by a value that is [less] than one
so this is not really truly a valid PdF for probability distribution function or
probability Model because I
Cannot list off. I cannot list off all of the
actual events and
The probabilities kind of go off and definitely. I mean [mike] in theory. I could just keep going and going to going forever in theory
And never have my first success, so the partial sum of this
You know attempted PDf table will always be less than zero the infinite sum is going to equal zero
equals zero
Less than one let's try that again the partial sum will always be less than one
only the infinite sum would actually equal one and a
Histogram computer make a histogram, and this case would be a relative frequency and zero [percent] if [you're] to construct a histogram
Out of these values [would] always be a white skewed distribution
You can see these high values on the left, and then they're dropping the dropping and drop as we go off
So kind of you know I would say that we're setting up a PDf table
But we've got these um [Deatha] thoughts here and really in theory goes on indefinitely
Well, what if we said that we were [only] willing to go up to seven trials [to] until we get our first success
well now x
here is
Going to be defined as the number of successes. They should have wrote that definition down
And I just [send] that incorrectly x is going to be the number of trials, okay?
what if you are only willing to make up to seven attempts to try and
get your first success well x now cannot be labeled as the number of trials until
I [give] my first success because I've you know written in this example that we're just going to give up after seven
So x is going to not [be] the number of trials for your first success, but simply the number of trials taken
Well all these numbers are going to match my previous PDf table up here
Except [for] the last one which now jumps from point O six for seven trials to point three eight four
Well that's because the problem states that we're going to attempt to get our first success
but we're not going to try after seven we're just going to give up well this now is no longer a
Geometric setting because well it starts off the internet geometric
We're going to go into our first success as soon as you say
We're going to quit after seven the whole test of the [VM]
Geometric falls apart you're not going until you are not going until your first success
You're going until you get to seven if you don't get your first success by then we're going to go ahead and quit
So this is no longer a geometric set and it's also not a binomial setting
Again, and was not fixed. [I] mean if we do get our first success on the first trial
we're going to quit, so this is a
probability distribution model for x to find this number of trials the designer
Geometric nor is it binomial all right page three
No, no, no no no no no
Okay, last bit of notes for geometric setting how do you find the mean and the standard [deviation] of a geometric setting?
Well the formula for the mean of x is 1 over p and the standard deviation for geometric variable is
the Square Root of 1 minus p
divided by P squared
now a
little bit funny about the Standard deviation formula
and it's just an intro to statistics book that I'm dealing with and I'm helping you guys with so
You know we just said that geometric distributions are always right skewed, so this is kind of like an oddball standard deviation formula
You're not Gonna
Have a geometric setting that is going to be normal [so] [just] be aware that the probability that it takes more than n
Trials before you see your first success
so
instead of like a previous example where I went up to my so I had to go to my
Seventh trial to see my first success
if I learned to know the probability that took more than
Seven trials until my first success there is a formula for that 1 minus p
Raised to the [Nth] power. I do not believe no, this is not only the
formula sheet, so if you want to use that formula on a test, you'll have to have its memory and
A couple more [examples] call it a day all right the expected number of trials to achieve your first success
Don't forget expected number of trials is another catchphrase for me. So the mean of x is 1 divided by 0.15
So again previous example [15%] success
we should be expecting us to have or have to take an average of six-Point-six seven trials and
so we see that first success on our last trial the standard deviation for that piece of
Data is six point one five and it's in a last [example] again
using your calculators distribution menu you can find these commands of
Geometric CdF
CdF cumulative
Distribution function, that's the same as by known
Cdf if you just watch my video [about] a binomial distribution or binomial setting and again
It's only four counts, and it only counts down if you're using an 83 or Ti-84 calculator
Texas instrument
So the probability that x is less than or equal to 7 is geometric cdF
the probability of
success and the number of trials that we're going until we see [are] per success comes out to be point six eight and
If I want the probability that takes more [than] seven trials
well more than seven the complement of more than seven is seven or less so I can take my
previous answer and subtract it from one and get 0.32 or if
You happen to remember this formula again, not happen if you [studied], but if you do remember this formula, remember
It's not going to be on your [Ap] stats formula sheet
You can use it if you remember you can use the probability that x is greater than seven is equal to one minus
the probability of success raised to Seventh power and
You are getting it the course answer of 0.32, so I'm mr.. [tarrou]. That's the geometric setting. Thank you. Go to your home