VCE Methods Question Thread!

[soNasty]

Oh and I need help with this too:
a coin is biased in such way that pr(heads)=3xpr(tails) find the probability that exactly six tosses out of ten land heads

[keltingmeith]

For the first question, you'd do it like you would anything else - you look at the probability distribution, and see what appears the most/what adds above 0.5.

If you have a TI-nspire, entering binomPdf(n,p) will give you the full distribution for a random variable X if . You can then add each element to find your median, and physically see what the mode is. Remember - going across, you start from X=0. If you're unsure you have the right number, you can put in binomPdf(n,p,X) to double check.

For the second question:
We know that a single toss represents a single distribution, and that the sum of probabilities must equal to one. This gives us . We also know that , which gives us . This also means that probability of heads is three times this, or . (you also could've done .

Now, we want to know what the probability of getting six heads is if we toss this particular coin ten times. We could consider each individual probability, but that's a lot of tree diagrams. However, you should notice that this is an example of a binomial probability with 10 trials and either 1/4 or 3/4 probability of success. Let's take heads to be a success, so our probability is 3/4. So, we now have . So, inputting this into the calculator for , we get 0.146.

[soNasty]

Thanks so much!

[juzza12]

Hey guys I'm not sure about finding the mean when it comes to probability density functions when you need evaluate an integral as x approaches infinity. The essentials textbook does include steps during these questions where you let k equal the limit as x approaches infinity but this does take time so I was wondering if this step was actually necessary for exams? I can do the question fine but it's just the working out steps that I'm not sure about. For example a question such as find E(x) for f(x)=1/(x^2) , x>1

[keltingmeith]

The way Essentials does it is just a way that allows you to evaluate the integral with the methods you know how to while still allowing you to be mathematically correct. You can, however, skip a couple of the steps:



At this point, you are still mathematically correct, as the points on the integral are limits. However, the moment you do this:



You're wrong, because you can't just treat infinity like a number - it is a concept. So, instead, you can just skip that step and do this:



Evaluating those two numbers in your head instantly, instead of taking the median step. If you still write it out like that, the assessors should see that you've computed the limits, and you haven't done anything wrong or said anything that's mathematically incorrect.

(the example I used was proving that is a probability density function)

[soNasty]

On the way to work Philomena must pass through six sets of traffic lights. The lights are independent of eachother annd the probability that Philomena must stop at any particular set is 0.7. Find the probability that Philomena stops at:

c) The first three sets of lights
d) The second and sixth sets online
e) Four sets of lights including the first two

could someone show me how to work this out using calc or by hand? I know it's a binomial Q.

[HEN_iP]

A function has rule of the form f(x)= |log_e(ax+b)|. It is known that f(0)=2 and f(1) = 1. Find the possible values of a and b

[knightrider]

Does anyone know how to find the asymptotes on a casio classpad calculator and does anyone also know how to complete the square on the claculator

[keltingmeith]

On the way to work Philomena must pass through six sets of traffic lights. The lights are independent of eachother annd the probability that Philomena must stop at any particular set is 0.7. Find the probability that Philomena stops at:

c) The first three sets of lights
d) The second and sixth sets online
e) Four sets of lights including the first two

could someone show me how to work this out using calc or by hand? I know it's a binomial Q.

c) The trick here is that you're going to see it and assume you have a binomial case - but you don't. See, if , then Pr(X=3) is going to give you the probability that Philomena stops at ANY 3 lights, not JUST the first three. Truth is, we don't care what she does at the last three lights, so the . You can test this by checking all the different permutations of this scenario,


d) I'm not quite sure what they're going for by "online" - assuming they mean she's fine between 2 and 6, what you want is Pr(?YNNNNY) - the reason there's a question mark in the first one is that we don't care if she made it or not, just what she did between 2 and 6. So, once again, we can just ignore that, and we get . Once again, you can double check this with all the possible scenarios.

e) Now, this is where things get interesting - we want to know the probability of her stopping at ANY FOUR. This instantly means we can use binomial stuff to find the probability. But, we also know that she's already stopped at the FIRST TWO. So, this means we get a conditional probability, . Because of this new arrangement, we have to use binomPdf very carefully. If you want to be safe, you can write out ever scenario again [Pr(YYYYNN) + Pr(YYYNYN) + Pr(YYYNNY) + Pr(YYNYYN) + Pr(YYNYNY) + Pr(YYNNYY)], and find all of them, or you can be tricky like this:

If we limit our binomial probability to just the last four events, we know that any two of those she must stop. This means we now have a binomial probability with 4 trials and probability of 0.7. Then, if we add on the from the first two she ran, we'll get the correct probability.

So, calculating it whatever way you want, you'll get 0.1297, giving us . Now, remember from before that the probability of stopping at the first two is just (as we ignore the other outcomes), so our final fraction becomes

[soNasty]

Thanks for all that!
Oops it was supposed to be "of lights" instead of "online" lol

The answer for d) is 0.0039 and e) is just 0.1297 (WHICH YOU HAD IN YOUR SOLUTION!) though!

[keltingmeith]

A function has rule of the form f(x)= |log_e(ax+b)|. It is known that f(0)=2 and f(1) = 1. Find the possible values of a and b

f(0)=2 and f(1)=1, this means that:

However, recall the definition of the modulus, meaning that for the first equation we have:

Using this, we then get


We can now use this to find the answer for a - but once again, that modulus is going to prove annoying, so watch carefully:


This means that b could take up to any of 2 values, and a could take any of up to 4 values (depending on what b is).

[keltingmeith]

Thanks for all that!
Oops it was supposed to be "of lights" instead of "online" lol

The answer for d) is 0.0039 and e) is just 0.1297 (WHICH YOU HAD IN YOUR SOLUTION!) though!

In that case, I'm not quite sure what they're doing for d, but it seems they've taken that she only stops at the second and sixth, and makes it through everything else (which is not implied at all...)

For e, I'd argue that since you already know that two of those are going to be the first it's actually a case of a conditional probability, but the book seems very content to disagree with me there, and that finding is enough. I'd ask your teacher on their take of it, though. (and if any other AN user sees this and wants to put their two cents in, feel free!)

[nerdgasm]

Hmm, I'm guessing that for part e), the reason why it's not considered a case of "conditional probability" by the textbook is because they've interpreted the statement "What is the probability that Philomena stops at four sets of lights including the first two" to mean "What is the probability that Philomena stops at four sets of lights AND two of those four happen to be the first two sets of lights", which is why taking the intersection appears to be sufficient (as opposed to, "What is the probability that Philomena stops at four sets of lights GIVEN two of those four happen to be the first two sets of lights", where you would probably treat it as conditional probability).

Personally, I don't really have a problem with the textbook's interpretation of that sentence - it seems like the added 'condition' (groan!) of having two of the lights Philomena stops at being the first two traffic lights is more of an additional constraint, rather than something that would limit the sample space. This is obviously just my opinion though.

[LiquidPaperz]

help please

[e^1]

[removed, see post below]

The way Essentials does it is just a way that allows you to evaluate the integral with the methods you know how to while still allowing you to be mathematically correct. You can, however, skip a couple of the steps:



At this point, you are still mathematically correct, as the points on the integral are limits. However, the moment you do this: ...

Just to add a tiny bit, when I did methods I wrote it like this, being aware (as Euler had said) that infinity is a concept. This however is not necessary.



edit: fix and error