VCE Methods Question Thread!

[Zealous]

Proportion - Do we write is as a percentage or a decimal? and how would we word it if it were a decimal?

Percentage is easier: In the long run, Ray will take the train on 68% of days. But if it were a decimal value?

[BasicAcid]

Proportion - Do we write is as a percentage or a decimal? and how would we word it if it were a decimal?

Percentage is easier: In the long run, Ray will take the train on 68% of days. But if it were a decimal value?

Unless it specifically asks for percentage, you leave it in decimals. Doesn't make as much sense, but it's the "more correct" way.

I did a VCAA paper (somewhere from 2006-2008 iirc) that asked for the proportion and the assessor's comments had the answer as a decimal.


In the long run, Ray will take the train 0.68 of the time. It's more about the maths than the wording.

[kevinnguyen]

Does anyone know how the probabilty inequalities work. e.g. pr(x=3) given pr(x>1)?

[lzxnl]

So...you want to work out Pr(x=3|x>1)
From the multiplication rule, this is equal to Pr(x=3 U x>1)/Pr(x>1)
But Pr(x=3 U x>1)=Pr(x=3) as the event x=3 is a subset of the event x>1 (aka if x=3, x must be larger than 1, so imposing the restriction x>1 makes no difference). Therefore, your probability is Pr(x=3)/Pr(x>1)
Now just refer to whatever data you have to work these out.

[kevinnguyen]

Does mean that since x>1 and x=3 is subset of it we use x=3?

[TrueTears]

Does anyone know how the probabilty inequalities work. e.g. pr(x=3) given pr(x>1)?

In classical probability theory, there are a few very very famous inequalities.

1. Chebyshev's (Tchebysheff's) inequality: for ANY random variable with mean and s.d . I'll let you see why this inequality is so famous (and powerful...)

2. Markov inequality: for any non-negative random variable X with expectation E(X). Exercise: try prove this from 1 (hint: consider the variance as a random variable...)

3.  Gaussian inequality:

There are literally hundreds more... however the majority of them can not be stated (at least "accurately") using classical probability, and this is where the power of measure theory comes in.

Both 1. 2. and 3. have even more formal definitions under measure theoretic probability (something which I urge students to have a read if you're considering a path down probability).

[kevinnguyen]

Does anyone know how to plot hybird function on a cas? e.g 1/1000(x-20),20<x<40  1/4000(120-x), 40<x<120

[Alwin]

Does anyone know how to plot hybird function on a cas? e.g 1/1000(x-20),20<x<40  1/4000(120-x), 40<x<120

on the casio you can just plot both separately and include the domain, eg in graphs and tables:
y₁= ____ | 20<x<40
y₂= ____ | 40<x<50
y₃= ____ | 50<x<80

EDIT: just remembered, there is a plot piecewise function,


hope it helps

[Zealous]

Does anyone know how to plot hybird function on a cas? e.g 1/1000(x-20),20<x<40  1/4000(120-x), 40<x<120

... and for TiNspire Users:


Similar to what Alwin just posted.

The open bracket (to enter two functions) can be found directly to the right of the 9 number (CAS CX), the given that symbol is with the inequalities.

Then from there you can plot f(x) in graphing (or you could jump straight into the graph page and put the hybrid function straight in).

[IndefatigableLover]

Does anyone know how to plot hybird function on a cas? e.g 1/1000(x-20),20<x<40  1/4000(120-x), 40<x<120

So just open up a new Graph page on your CAS. The click on the button that's bottom left of the 'Del' sign and find the hybrid function. Basically include your values for the two separate lines which your hybrid function gives you and click enter (remember to put in domains). Afterwards, adjust your window settings (x: Min - 0, Max - 100. y: Min - 0, Max - 0.05) and you should be able to view it in a graph form



(Not sure if I interpreted your question right LOL but hope it helps (Y))

[clıppy]

If it's just plotting don't forget you can do the bits separately.
On a graph page, let f1(x) = the first equation, restricted
let f2(x) = the second equation, restricted

If it's the shape you're looking, for this can be quicker and you can still determine y-values using Menu-5-1 and choosing whether you are working with f1(x) or f2(x)

[Sanguinne]

The minimum number of times that a fair coin can be tossed so that the probability of obtaining a head on each trial is less than 0.0005 is

I know the first step is  0.5^n < 0.0005 but I don't know why?

[SocialRhubarb]

The chance of a head being thrown on each toss is 0.5.

If we were working out the chance of 3 heads being thrown from 3 tosses, we'd need a head and then a head and then a head, the probability of which is

Similarly, the probability of throwing 'n' heads from 'n' throws is , and we want this number to be less than 0.0005.

[TrueTears]

The minimum number of times that a fair coin can be tossed so that the probability of obtaining a head on each trial is less than 0.0005 is

I know the first step is  0.5^n < 0.0005 but I don't know why?

There are many ways to go about this depending on what framework of probability you're working within.

1. Classical: You can approximate the binomial event as following a normal distribution as per the central limit theorem, then simply use the cumulative inverse normal function to work out n.

2. Bayesian: This paradigm of probability is completely different from classical. Here, we assume n is some random variable (rather than as a parameter in the classical case...). Hence, we can specify some kind of distributional assumption for n. In the bayesian setup, we have the famous identity . In words, this translates to "Posterior is proportional to the likelihood function times the prior". The prior in this case is some initial "guess" you have of the distribution of n, the likelihood function then "updates" your views based on some kind of information/experiment, in this case, it'd be the binomial coin tossing experiment. Finally, the posterior is a combination of your views supported by your initial guess and the data (experiment). The rest is follows from classical probability but conducting all inferences on n (as it is a random variable NOT a parameter). So you can't really find an unique value of n, all values of n will be entirely categorized as probabilistic values rather than a definitive value.


[1.] http://en.wikipedia.org/wiki/Classical_definition_of_probability
[2.] http://en.wikipedia.org/wiki/Bayesian_probability#Bayesian_methodology

[Hancock]

There are many ways to go about this depending on what framework of probability you're working within.

1. Classical: You can approximate the binomial event as following a normal distribution as per the central limit theorem, then simply use the cumulative inverse normal function to work out n.

2. Bayesian: This paradigm of probability is completely different from classical. Here, we assume n is some random variable (rather than as a parameter in the classical case...). Hence, we can specify some kind of distributional assumption for n. In the bayesian setup, we have the famous identity . In words, this translates to "Posterior is proportional to the likelihood function times the prior". The prior in this case is some initial "guess" you have of the distribution of n, the likelihood function then "updates" your views based on some kind of information/experiment, in this case, it'd be the binomial coin tossing experiment. Finally, the posterior is a combination of your views supported by your initial guess and the data (experiment). The rest is follows from classical probability but conducting all inferences on n (as it is a random variable NOT a parameter). So you can't really find an unique value of n, all values of n will be entirely categorized as probabilistic values rather than a definitive value.


[1.] http://en.wikipedia.org/wiki/Classical_definition_of_probability
[2.] http://en.wikipedia.org/wiki/Bayesian_probability#Bayesian_methodology

TT, you're aware that this is a Year 12 Methods thread and that no high school students understand anything you're saying yeah?