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March 19, 2024, 03:36:58 pm

Author Topic: VCE Methods Question Thread!  (Read 4795802 times)  Share 

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keltingmeith

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Re: VCE Methods Question Thread!
« Reply #18600 on: July 05, 2020, 04:04:30 pm »
+7
Hello!

I have a question to ask on binomial distribution: What is the least number of times a fair die should be rolled in order to ensure that: a the probability of observing at least one 6 is more than 0.9 b the probability of observing more than one 6 is more than 0.9?

How do I use the formula Pr(X=x) = nCr(n,x) p^x (1-p)^(n-x) in my approach to this question?
'n' has to stay unknown, but I'm not sure how to format the probability being more than 0.9 and the value of x being between 1 and n.

Help would be appreciated!!  :D


Okay, so I think for probability questions, it's often easier to have things make sense in your head BEFORE you worry about playing with equations. So, let's start by looking at a:

the probability of observing at least one 6 is more than 0.9

Okay, so we have the probability > 0.9
We want AT LEAST one 6. Here's the thing - working with "at least" is really hard, particularly when we don't know how many trials we have. But, we DO know from the law of total probability that the probability of getting "at least one 6" is the same as 1-P("at least one 6"s complement). Or, 1-P(no 6). So, this gives us:

P(at least one 6) > 0.9
1 - P(no 6) > 0.9
P(no 6) < 0.1

Okay, so what's the binomial distribution look like here? We don't know what n is... But, we do know what p is - it's the probability of either not getting a 6 (5/6) or the probability of getting a 6 (1/6). Since we're interested in NOT getting a 6 in our new equation, why don't we call the "success" not getting a 6 (5/6). So, this gives us the equation:



Alright, so what's x? Well, x is the amount of times we DON'T roll a 6, and we DON'T want to roll a 6. It can get a little confusing with the more abstract "n" in there, so let's try using some actual numbers to try and understand what's going on:

If I roll a die 5 times, how many times do I want to NOT get a 6? 5 times
If I roll a die 3 times, how many times do I want to NOT get a 6? 3 times
So, if I roll a die n times, how many times do I want to NOT get a 6? Hopefully you see the pattern - and if you don't, please ask for us to try and explain again. But yes, it would be n times

So, now our equation is:




Which can now be solved with the use of some logarithms. However, you could also make this easier on yourself, and just pick values of n until (5/6)^n is greater than 0.1 - I'm getting that n=13 by using this method on my calculator, and it only took me half a minute to do ;) Of course, if you're convinced to doing it the method with logarithms:






Remember that log(1)=0 AND is strictly increasing for all numbers it's defined for, so log(<1)<0, which is why I needed to swap the sign. And, since the first integer (since n HAS to be a positive integer) greater than 12.6 is 13, the answer is 13. Also, the choice of base doesn't matter - what's important is that your calculator can do the maths to that logarithm. I used base 10 because it made my life easier as I no longer own a CAS calculator or program, and so had to use excel to calculate it for me

You should be able to apply this thinking to part b, so use that part to test your understanding. If you're still struggling, let us know what you're having problems with, and we'll take it from there

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Re: VCE Methods Question Thread!
« Reply #18601 on: July 05, 2020, 04:21:57 pm »
+3
Hello!

I have a question to ask on binomial distribution: What is the least number of times a fair die should be rolled in order to ensure that: a the probability of observing at least one 6 is more than 0.9 b the probability of observing more than one 6 is more than 0.9?

How do I use the formula Pr(X=x) = nCr(n,x) p^x (1-p)^(n-x) in my approach to this question?
'n' has to stay unknown, but I'm not sure how to format the probability being more than 0.9 and the value of x being between 1 and n.

Help would be appreciated!!  :D
I would do guess or check.
Alternatively you could, graph/tabulate the result that's pretty quick, do you know how to find the greatest term in a binomial expansion by setting up inequality? that may help.

EDIT: Just read what keltingmeith did, I didn't see that he replied.
« Last Edit: July 05, 2020, 04:25:38 pm by 1729 »

The Cat In The Hat

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Re: VCE Methods Question Thread!
« Reply #18602 on: July 10, 2020, 12:32:08 pm »
+1
For my bound book I want to include when to use radians and when degrees on the CAS, because I'm always scared I'll get it wrong. As you may see I don't really understand them... anyway, when do I use them? I mean apart from circular functions, what do I use, because I guess circular functions (please correct me if I'm wrong...) if it has π in it it's a radian measure and if it doesn't it's degrees? Is that right?
Thanks
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Evolio

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Re: VCE Methods Question Thread!
« Reply #18603 on: July 10, 2020, 12:38:04 pm »
+3
Hello.

You may also use them when calculating the angle that the gradient of the tangent makes with the x axis (in differentiation), or the inverse, where you need to calculate the gradient of the tangent using the angle given. This concept was in one of the recent exams.
Yes, that is correct. Also, it'll be clear when they give it in degrees as there will be the degrees sign so that should be helpful to identify which is which
« Last Edit: July 10, 2020, 12:49:07 pm by Evolio »

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Re: VCE Methods Question Thread!
« Reply #18604 on: July 10, 2020, 12:40:02 pm »
+2
Hello.

You may also use them when calculating the angle that the gradient of the tangent makes with the x axis (in differentiation), or the inverse, where you need to calculate the gradient of the tangent using the angle given. This concept was in one of the recent exams.
Yes, that is correct. Also, it'll be clear when they give it in degree as there will be the degrees sign so that should be helpful to identify which is which
Thanks for the quick reply! :) Clear and concise. Thanks!
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Re: VCE Methods Question Thread!
« Reply #18605 on: July 10, 2020, 01:33:30 pm »
+2
if it has π in it it's a radian measure and if it doesn't it's degrees? Is that right?
Thanks

Not necessarily. sin(90) ≠ sin(90°).

Evolio's approach is probably best: any angle measured in degrees will have the "°" symbol; otherwise, it's measured in radians.

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Re: VCE Methods Question Thread!
« Reply #18606 on: July 10, 2020, 02:43:32 pm »
+1
Thanks!
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Re: VCE Methods Question Thread!
« Reply #18607 on: July 11, 2020, 11:29:32 am »
+1
Hello all :)

I'm struggling with this question, what are the steps to solving this?


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Re: VCE Methods Question Thread!
« Reply #18608 on: July 11, 2020, 11:59:48 am »
+2
Hello all :)

I'm struggling with this question, what are the steps to solving this?
You won't actually get those on the exam as absolute values are off the study design.

But to do this sort of question, just calculate the area below the graph for the interval specified. You can draw the graph first if it helps you visualise what it would look like. :)

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Re: VCE Methods Question Thread!
« Reply #18609 on: July 11, 2020, 12:20:36 pm »
+5
Hello all :)

I'm struggling with this question, what are the steps to solving this?
You want to find the area under the curve until 1.5, first visualize the graph. Technically It would be an integral but since |1-x| is a triangle you can use triangle formula.


Do you understand how we get this weird graph looking thing?
You first graph the piecewise function as it says, then you want to find the area under it from x = negative infinity to x = 1.5.

From there you can do:

(0.5 * 1 * 1)+(0.5 * 0.5 * 0.5)
0.5 + 0.125
0.625

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Re: VCE Methods Question Thread!
« Reply #18610 on: July 11, 2020, 10:37:55 pm »
0
Hi everyone!

I'm not really sure how to even begin working this question out since I always suck at questions that has only just variable letters *sigh*.

"For f(x)= 3ln(x+x/2),
- If f(u-2) + f(v-2)=f(auv+b) where u and v are positive real numbers, find the values of a and b.

- For what values of u does f(u)+f(-u)=f((u^2)/2) hold?"

Any help would be greatly appreciated! Do I start with equating equations or summ?
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keltingmeith

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Re: VCE Methods Question Thread!
« Reply #18611 on: July 11, 2020, 10:42:46 pm »
+2
Hi everyone!

I'm not really sure how to even begin working this question out since I always suck at questions that has only just variable letters *sigh*.

"For f(x)= 3ln(x+x/2),
- If f(u-2) + f(v-2)=f(auv+b) where u and v are positive real numbers, find the values of a and b.

- For what values of u does f(u)+f(-u)=f((u^2)/2) hold?"

Any help would be greatly appreciated! Do I start with equating equations or summ?


I think the easiest way to approach these questions is to make them look less scary. Right now, you've got some alphabet soup and some really scary and complicated looking equations. So let's break it down and ask - what do you need to know?

Well, firstly, you need to know what f(u-2) is. Then, you need to know what f(v-2) is. Then, you need to find what f(auv+b) is.  Once you've got all of those, then you can think about what f(u-2)+f(v-2) is, and how that looks similar to f(auv+b).

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Re: VCE Methods Question Thread!
« Reply #18612 on: July 12, 2020, 08:46:32 pm »
0
Hello  :)

I am having trouble with this question, particularly with integrating f(x)

keltingmeith

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Re: VCE Methods Question Thread!
« Reply #18613 on: July 13, 2020, 12:18:59 am »
+2
Hello  :)

I am having trouble with this question, particularly with integrating f(x)

Remember that integration is basically reverse differentiation. So, if you differentiate x^2 and get 2x, then if you integrate 2x, you'll get x^2. What can you differentiate to get cos(x)?

It's also worth adding that random variables are scrubbed from the study design this year, so you won't be asked a question like this anyway (though being able to do the integrals can still be expected, you'll just see them in different circumstances)

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Re: VCE Methods Question Thread!
« Reply #18614 on: July 13, 2020, 10:10:45 am »
0
Hi everyone,
I would appreciate some help with the following Year 11 Math Methods question:

sin x = 0.3, cos a = 0.6 and tan b = 0.7

Find tan ((pi/2) - b).