Hello and welcome to Mr Tompkins Ed Tech and the next video in our series looking at solving
IB and A level exam questions using the Casio Classwiz FX-991EX Calculator. We are looking at
the binomial distribution today, which you will find on the old S1 Statistics and Probability
paper and also the new 2017 specification AS and A level.
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Back to the Maths, today we'll be taking a look at the Binomial distribution and how
we can use the Casio Classwiz calculator to generate both single and cumulative probability
values.
Often in exam questions you will need to figure out for yourself that the binomial distribution
needs to be applied. You're looking for scenarios involving a series of independent trials,
each trial with a constant probability of success (which we normally denote with p),
and where there are two distinct outcomes, “success” of “failure”. Importantly,
there should be a fixed number of trials, denoted by n. This distinguishes it from the
geometric distribution, where you instead continue with the trials until you are successful
and then stop. If you are asked to justify why a problem
can be modelled as a binomial distribution, it is important to rewrite these conditions
with a clear reference to the context of the problem, and not just rattle them off, or
you run the risk of not being given full credit for your answers.
However, this particular question is very straight forward - there is no context for
us to interpret and it tells us up front that it's a binomial distribution. We just need
to tap the parameters into the calculator and write down the results. Now if you didn’t
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.
The parameters for this distribution in part i) are number of trials n = 8 and probability
of success p = 0.55 and part 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 it can take the values 0 through 6. So the probability of X less than 7, is
equal to the probability of X less than or equal to 6.
First up, lets put the calculator in Distribution mode by pressing "MENU" and then “7”.
You can see tha the Classwiz can produce values for the Normal, Binomial and Poisson distributions.
We will look at the others in future videos but today we are going to be focusing just
on the binomial distribution. Now the Classwiz can generate either individual
or cumulative probabilities and can do this for either a single value or for a list of
values. The “List” option is useful if you have a series of question based on the
same distribution, like we do in the first part of this question.
So for this one, we are going to choose the binomial cumulative distribution. We're being
asked to find quite a few different probabilities from this distribution so lets choose the
“List” option. First of all, I'm going to tap in all the possible values that our
random variable X can take, 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.
Pressing “=” takes you to the next row and pressing it again at the end of the list
takes us to this screen where we can enter our parameters n and p. The number of trials
n is 8, and the probability of success p is 0.55. Pressing “=” again takes us back
to the table view, but now we have the second column populated with the cumulative probabilities
displayed to 4 decimal places. You can use the direction keys to move around the table,
and the selected value is shown to a higher degree of accuracy underneath. We want X less
than or equal to 6. Just be mindful that this first column refers to the row number, which
is a little bit confusing actually, and might lead to errors when under exam pressure. So we are
looking up 6 in this middle column and then across, which is 0.9368, or 0.937 to 3 significant figures.
For part b) we want the probability that X = 5,
so we need to switch from the cumulative probabilities to the single binomial
probability distribution. Press “OPTN”, “1” followed by “4” and then “1”.
Our values of X from 0 through 8 are still here, but the probabilities have disappeared.
Pressing “=”, you’ll see that the values for n and p are still here from last time,
so just press “=” again, and the probabilities will be populated into the table. Scrolling
down to 5 and again making sure we read across from the value of X and not the row number,
we find that the probability that X=5 is 0.2568, which we'll round to 0.257 to 3 significant
figures. Part c) is a little more complicated, and
again it helps to visualise this by listing the possible values of X and the ringing the ones we want. 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 probabilities 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. Either approach is fine, but I normally go for the one that requires the
least number of values to work with, so I’m going to switch back to cumulative probabilities
– by pressing “OPTN”, “1”, “DOWN”, “1”, “1”, and pressing “=” to
regenerate the probabilities. P(X<5) is 0.7798 subtract P(X<2) 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 distribution but with different parameters. We only require a single
probability this time, so there is no advantage in generating the whole table of results likes
we did in the first part of the question. Pressing “OPTN”, “1”, “4” and
this time “2” for "variable" takes us directly to the parameter input screen, and this time
there is a space for an additional value for x, or in this case y.
The value of Y i'm looking for is 2 so we enter it here, n is 10, and p is 5/12ths,
so I’ll enter 5 divide by 12. 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
to “Calculate” mode by pressing “Menu” “1”, the variance of Y here is npq so
10 x 5/12 X 7/12, which is equal to 175 over 72.
You've now learnt how to use the distribution function on your casio classwiz to generate
single or cumulative binomial probability 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.
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