Hello and welcome to Mr Tompkins Ed Tech and the next video in our series looking at solving
IB and A level S1 Statistics and probability exam questions using the Casio Graphical Calculator.
I'm using the Fx-9860 here but don't worry which model you have. Casio calculators have
not really changed that much in terms of their basic functionality over the past 12 years,
so whether you have a shiny new CG50, a dusty old FX9750, or any of the models inbetween,
you can still join in.
In previous videos, we've covered standard deviation, Pearson's PMCC, and least squares
regression lines. So enough of the stats already, time for a bit of probability. Today we'll
be taking a look at the Binomial distribution and how we can use the Casio Graphical calculator
to generate both single and cumulative probability values.
So over to the exam question. This one is very straight forward - there is no context
for us to interpret and it tells us up front that it's a binomial distribution. Often the
examiners are less kind and you'll need to work out yourself that the binomial distribution
needs to be applied. In these cases, you're looking for scenarios involving a fixed number
of trials and an equal chance of success for each trial.
But nothing like that to worry about here. We just need to tap the parameters into the
calculator and write down the results.
Now if you didnt have a calculator like this at your disposal, you would need to be using
the somewhat archaic mathematical tables supplied in the formula booklet. But lucky for you,
you dont have to. All the values in these tables can be calculated in an instant on
your Casio device.
First up, lets put the calculator in STATS mode by pressing "2", and clear out any existing
data.
I'm going to show you two different methods for generating the probabilities, depending
on whether you want the whole distribution or just one particular value from it.
The parameters for this distribution are number of trials n = 8 and probability of success
p = 0.55. We're being asked to find quite a few different probabilities here, so I'll
use this one to demonstrate how we can generate the entire distribution. It's arguably a bit
overkill here, but a useful technique nonetheless. First of all, I'm going to tap in all the possible
values for our random variable X, which in this case is the numbers 0 through 8. We wont
actually need all these, but it's quicker to type them all in than think about what
I do and dont need right now.
OK, now I've entered these, I can get the calculator to generate the associated probabilities.
To do this, press "F5" for the distribution functions. You can see that the calculator
can generate quite a few types of disributions, but we need "F5" the binomial distribution
here. We are now presented with 3 further choices. Let's select "F1" for Binomial Probability
distribution first.
I need to configure this table using the distributions parameters and tell the calculator where to
output the values. First of all, it's asking where the values of X are coming from. I can
choose either a list or one particular value of X. I want to generate the entire distribution
so I'll pick list. Next I specify which list contains the input values, which is List1.
Next I enter the number of trials n, which is 8, and the probability of success p, which
is 0.55. "Save Results" refers to where want to post the output values. I'm going to tell
the calculator to post these to "list 2"
So lets "execute" and the calculator will show all the probabilities - take care here
because the numbers on the side start at 1 rather than 0 so dont match up with the values
of X. Better to look at the results in the data table, so press "EXIT" and "EXIT" to
go back to this.
We can see List 1 has our random variable values for X, and List 2 has all the associated
binomial probabilities. Before we start answering the question though, I also want to generate
the cumulative binomial probabilities too, as some of the questions are asking for ranges
of values.
I can do this by again pressing "F5" for distribution and "F5" For binomial distribution, but this
time we'll choose "F2" for cumulative proabilities. You'll notice that the setup table still contains
all the data settings from last time around, so I just need to change where the results
need to be saved to . IF you leave it as "List2", you'll overwrite the previous results, so
I'll change this to "list3", and execute. Again, better to EXIT, EXIT to get back to
the main data table before we attempt to analyse these results.
ok, so we now have the entire distribution, with values of X from 0 through 8 in list
1, the associated binomial probabilities in list 2, and the cumulative probabilities in
list 3. Now it was very quick to rattle all these off and we now have all the information
we need to answer any question the examiner might throw at us.
Question A asks for the probability that X is less than 7. When students make mistakes
with these questions, it's nearly always with picking the appropriate range. If you also
struggle with this, I think it's helpful to visualise this range by listing of possible
values and putting a ring around the ones we actually need. So X being less than 7 means
0 through 6. So the probability of X less than 7, is equal to the probability of X less
than or equal to 6, and that is given by our cumualtive probability next to 6 here: which
is 0.9368, or 0.937 to 3 signifcant figures.
Question B asks for the single probability when X = 5, which is our value in List 2 next
to the 5 - 0.2568, which we'll round to 0.257 to 3 significant figures.
Question C is a little more complicated, and again it helps to visualise this by listing
the possible values of X. Here X is less than 6 and greater than or equal to 3, so X can
be 3,4 or 5. So we can either add these 3 single probablilities together, or we can
use the cumulative probabilities and subtract the probability that X is less than or equal
to 2 from the probability that X is less than or equal to 5. I always go for the one that
requires the least number of values to work with, so 0.7798 subtract 0.0884, which gives
0.691 to 3 significant figures.
On to part 2, which concerns itself with a new random variable Y, which also happens
to follow a binomial distrubution but with different parameters, so we'll need to go
back and enter the distribution from scratch. So "F5" for the distibution function, and
F5 for binomial. We only require a single probability this time, so F1 for single binomial
probabilities, and because there is only one value to find, there is no advantage in generating
the whole table of results likes we did in the first part of the question. So rather
than inputing a range of values from a list, this time we'll input a single variable.
The value of Y i'm looking for is 2 so we enter it here, n is 10, and p is 5/12ths.
Now the maths tables wont have 5/12ths, which means that without your graphical calculator,
you would have to do this one using combinations, so 10 choose 2, p to the power 2, q to the
power 8, but here we can just tap in the parameters and let the calculator work it out and we
write it down - so 0.105 to 3 significant figures.
Variance of a random variable following a binomial distribution is given by npq. The
calculator wont do this for us strangely, so we'll have to type it in ourselves- Switching
the maths mode, the variance of Y here is npq so 10 x 5/12 X 7/12, which is equal to
175 over 72.
So thats all for this video. You've learnt how to use the distribution function on your
casio graphical caculator to to generate single or cumaltive binomial probabiility values,
or the entire distribution with the press of a few buttons, and no need to use the maths
tables from the formula book.
Please give this video a thumbs up if you found it useful, and leave a comment if you
have any questions or suggestions for future videos.
I'll be adding more videos in this series soon, so why not hit subscribe and you'll
be notified as soon as they appear.
Thanks for watching and hope see you next time.
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