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Statistics – Binomial Distribution w/TI-83/84 Calculator (Example 2)

October 21, 2019

This is Chris Ferbrache and I’m going
to review the binomial formula and we’re going to have an example with a two-part
example and this is a little bit more complex than the previous one which used
the probability density function in this case we’re going to have some questions
that say less than at least and what that means is we’re going to use our
cumulative density function which allows us to do one calculation on the
calculator and it’ll give us the result all the way up to the certain value that
we’re looking for okay we’ll get started okay so we want to use the binomial
formula to calculate the following probabilities for an experiment in which
n equals 6 and P equals 0.3 which is our success okay so we’ll start off with
listing out what we’re working with on the first one we’re going to do less
than lesson two so that’s what we’re working with so
less than two means we’re looking for zero and one so it has to be less than
two it essentially can’t touch two so less than two okay
and it looks like this okay so we have to find the probability for the zero and it ends up being six factorial 0 factorial 6 minus zero factorial and
then we do a point three zero parentheses and then 0.7 which is our Q
value and it’s six minus zero and when we work this out I’ll do this on this
one we won’t work it out I always recommend
calculating parts instead of trying to put it all into your calculator because
you’re bound to have a mistake with order of operations and things such as
that we end up with for the first one 720 over 720 and then one and then point
one one seven six and it goes on for a little bit before okay
and then our answer for this one ends up being point one one seven six so that’s
four zero and then we have to calculate the answer for one
and that is very similar six factorial and then 1 factorial 6 minus 1 factorial
and then we do a point 3 it should be 1 point 7 and then it should be 6-1 6-1
and our answer ends up being cut to the chase about point 3 0 to 5 ok so to
solve this what we would do is we’d add up these two figures we add up this
figure right here and add up this figure right there and when we do that we end
up with our grand total of approximately 0.4 to 0 1 and that is for our back out
that is for our less than 2 problem ok now let’s look at it on the calculator
I’m a calculator we do have an emulator so I’ll pull that up there emulator I’ll put it over here there emulator we
we need to turn on clear it okay we’re going to go to second and then D is TR
for distributions and this is where we’ll find our binome CDF cumulative
density function formula or a program within the calculator so we go up to
binome CDF in this case CDF we’re gonna use the cumulative one in the previous
example I did we used PDF which is not cumulative it finds a certain point okay
so I’m going to hit enter when I have binome CDF selected so we’re gonna put
in our trials in this case our trials are six six trials our p-value is 0.3
and our x value is where all the way up to now in this case it’s less than two
and because it’s less than two we’re not gonna put in two we’re gonna put them
one because it has to be less than two so it’s only zero and one now I hit
enter and it says pace because it’s gonna paste in into the application
formula our values now if your calculator isn’t really fancy with the
the input screen then you pull up the binomcdf and then you have an open
parenthesis and you put in six comma point 3 comma 1 and we hit enter and we
end up with point four two zero one essentially and our answer was moved
real quick our answer was about the same so 0.42 zero one actually identical okay
so that’s good let’s do the next one for the next one it gets a little bit
harder so for the next one we are going to find
the probability that X is at least four so we’re looking for four and more so
for four more we’re going to it’s going to be P equals X is greater than or
equal to four and that’s essentially what we’re looking for so we have P
which equals four five and six is what we’re doing for P 5 and P which is 6 and
when we work this out I’ll draw these out real quick we end up luckily I hope
this already calculated let’s see we end up with 6 on the top of all of them
expect oral six pectoral 6 factorial and then down below and we have four five
six so four five six and then it’s always six minus earning a space five
factorial and then let’s see it’s six minus six factorial oh and actually I
need put the factorials right here factorials in between the five right
there it’s gonna be 6 minus 4 factorial and I was going to be point three fourth
power point three to the fifth power and point three to the sixth power and
and multiplied by 0.7 which is our Q value which means it’s the failure part
of our success probability so it’s point 7 and we to use 6-4 and then it is point
7 and it’s six minus five and point seven six – six okay so we’ll just look
at the answers real quick so we calculate that all out like I said
before always see these in parts I recommend doing the bottom of this the
denominator and then the numerator and then the parts out here and then
multiply them all together it just makes more sense okay so our answer for the
individual parts of zero five nine five it’s a little messy point zero nine
right three point zero five nine and then point zero one zero two and then
point this is actually gave me a scientific notation on this one zero
zero seven two nine so it’s a very small probability okay
and when we add these all up we sum up the three we end up with point zero
seven seven zero for my papers moving so that’s our answer
point zero seven zero four so it’s about seven percent let’s say okay let’s do
this on the calculator and on the calculator and pull up the emulator we
have a couple tricks that we can do on the calculator we can do this and clear
it now I do second and distribution I go down you can actually go up and I’ll
bring you to the bottom of our options we go binome CDF cumulative density
function because we are working with multiple or
a range of values and we’re going to do trials which should most of the status
should be the same so six that’s great point three for success for our P and
that’s great now our x value now in this case there’s
a couple tricks we can do we are looking for the at least four so we’re looking
for four or five and six and there’s only six trials so what we can do is we
can find the cumulative probability of zero to three and we’re going to do that
so we find the cumulative probability of zero to three we do enter and we get
point point ninety to ninety nine five nine five so you go why are you doing
that well it’s easier sometimes to find the inverse when we do this with
cumulative probabilities because the cumulative probabilities always start
from the bottom go up so we’re gonna subtract this from one and when we
subtract it from one now think about this it’s it’s this is the calculation
what we just did was the calculation for zero one two three and all that’s left
is four five and six so it’s going to be a very small number but so we’re going
to subtract it from one so I can take this number this right here and I can do
one minus and then if I do second answer it’s going to load this value right here
or I could just go up and pull it down with my selecting it but that’s okay we
have right here and it gives us the answer of Wow
it’s almost exactly the same it’s exactly the same it is point zero seven
zero four it’s a little bit more exact for seven point zero seven zero for
seven point zero seven zero for seven which ends up being about all the way
over about seven percent so about seven percent there’s about a seven percent
probability that I’ll close this that if we are trying to find the probability
that X is at least four meaning four five and six that we have about seven
percent probability there’s about a almost 93 percent probability that it
will be less than four so that’s something to think about whenever the
probability is low we end up it’s very very hard to get when we get up into the
higher range it’s always very improbable and because the probability is 0.3 it’s
always going to be more probable more probable on the low end and that’s about
it. Thank you.

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