Poisson distribution

secretliker SG

Just part (a), I think my answer key is wrong.

In the manufacture of glass panels, small bubbles occur at random at an average of 1 bubble in every glass panel. The number of bubbles detected in a panel is denoted by X and follows a Poisson distribution.

(a) find the probability that, in a randomly chosen glass panel, there are at least three bubbles. [ans=0.0144?]

jiaxing2 SG

did u get 0.0803?

Uncertain SG

X - Po(1)

P(X >=3) = 1 - P(x=0) - P(x=1) - P(x=2)

use this formula to get 0.0803 (same as jiaxing)

secretliker SG

I got 0.0803 too. Thanks, now I can tell teacher the answer wrong.

Uncertain, do you press calculator 3 times to get P(X=0), P(X=1), P(X=2)?

Or can your calculator calculate directly P(X<=2)?

Uncertain SG

Nope, cos i am a FM student last time... speed in calculation meant something to us.

Ok enough of talking, i teach u a simple method.

for P(x=0) + P(x=1) + P(x=2), type e^(-1) [ 1+1+1/2] in your calculator

In general, P(x=0) + P(x=1) + P(x=2) + .... = e^(-y) [ 1 + y + y^2/2 + ....] , y = lumda

anyway, ur calcuator cannot calculate P(X<=2) directly.

secretliker SG

My calculator can calculate P(X<=2). Maybe last time cannot.

You're using the formula for Poisson distribution right?

( e^-y ) [ (y^x) / x! ] , where y=lambda.

It's in the formulae booklet but I seldom use it cos now everything is simplified by the GC.

Uncertain SG

My calculator can calculate P(X<=2). Maybe last time cannot.

You're using the formula for Poisson distribution right?

( e^-y ) [ (y^x) / x! ] , where y=lambda.

It's in the formulae booklet but I seldom use it cos now everything is simplified by the GC.

haha secretliker.

Let me share with u what i told my math student and what me and eagle share in common about the use of calculator.

I strongly discouraged u from relying on calculator as....

1) u will miss out a lot of concept when u rely on it. a classical example is the sketching of graph.... which involves differentiation, partial fraction and finding the intercepts.....

2) what if ur calculator spoil and cannot be replaced? U have to re-learn all the functions (worst still some functions got several symbols and wordings).

Therefore, i urge u to use calculator for PLAIN calculation and not relying on it to solve question (but for double checking is fine).

H2mathtutor SG

Hi Uncertain,

I certainly agree with you that it is still important to learn the concepts and not be totally reliant on the use of the GC.

The intent of introducing the GC into the A level curriculum (in my personal opinion) is to take away the "hassle" of mathematical calculations/manipulations, so that students can focus better in learning the concepts.

For secretliker's question, it makes more sense to use the GC than to apply the Poisson formula.

I taught in a JC for 3 years and have observed students who are not adaptable enough to tap on the powerfulness of the GC. One analogy I used to share with my students was: "If you can get to your destination faster in a sports car, why would you want to use a bicycle?"

Mikethm SG

haha secretliker.

Let me share with u what i told my math student and what me and eagle share in common about the use of calculator.

I strongly discouraged u from relying on calculator as....

1) u will miss out a lot of concept when u rely on it. a classical example is the sketching of graph.... which involves differentiation, partial fraction and finding the intercepts.....

2) what if ur calculator spoil and cannot be replaced? U have to re-learn all the functions (worst still some functions got several symbols and wordings).

Therefore, i urge u to use calculator for PLAIN calculation and not relying on it to solve question (but for double checking is fine).

I disagree with you. Why? Because the O level A Maths syllabus took away the topic formerly known as function (eg. f(x), g(x), f^-1(x) etc)... In addition, they also simplify the requirements for sketches of graphs of trigo, linear, quadratic, modulus etc...

Learning by the old school way would take something like 10hrs... and would equip the student to do sketch of stuffs like y = |3cos[(pi)x-2]+1|-5... the example I chosen is "illegal" (to be an AM question) for the 3 reasons

1. The coefficent of (pi)x is illegal, must be an integer.

2. Cannot have +/- inside the ratio.

3. Cannot have anything outside the modulus.

You may feel that the syllabus is "wrong" but think about it... do you really need to sketch? Cannot draw/plot meh? Especially in this age and time when computers can do the "plotting" for you anytime... I don't know about you but if my GC is down,

1) I can use my PC...

2)my PC is down... I can use my laptop... my laptop is down...

3)I can use my MacBook... my MacBook is down...

4)I can use my spare PC... my spare PC is down...

5)I can use my webserver PC... I webserver PC is down...

6)I can use my LG KT610 handphone... my KT610 down...

7)I can use my Nokia with java applications...

Tell me what are the chances that I do not have access to a piece of equipment which can draw me a graph?

I agree with the change in emphasis personally. I think the syllabus reviewers did an excellent job phasing out unnecessary skills to make room for more meaningful stuff such as Geometric Proof. Complicated graph sketching is just about mechanic skills whereas Geometric Proof build up the ability to reason and argue mathematically.

Concepts over technicalities. Kudos to the Maths Specialists in the MOE.

eagle SG

1) I can use my PC...

2)my PC is down... I can use my laptop... my laptop is down...

3)I can use my MacBook... my MacBook is down...

4)I can use my spare PC... my spare PC is down...

5)I can use my webserver PC... I webserver PC is down...

6)I can use my LG KT610 handphone... my KT610 down...

7)I can use my Nokia with java applications...

Tell me what are the chances that I do not have access to a piece of equipment which can draw me a graph?

First simple question.

During exams (might be A levels), your GC is suddenly down, and no one else have a spare since it is so expensive. Then how?

Second simple question.

Without knowing properly the concepts beforehand, how did programmers even come up with the most efficient software?

The current A level syllabus isn't even concept over technicality. Many of the students do not even know how to sketch a complicated graph from scratch; they only know how to plot it on the GC. This is in my opinion, technicality (of how to use the calculator) over concepts.

Sure, computers can do many things, but to me, it is still wise not to be overly reliant on them, which is actually what Uncertain and I meant. :D

Mikethm SG

First simple question.

During exams (might be A levels), your GC is suddenly down, and no one else have a spare since it is so expensive. Then how?

Second simple question.

Without knowing properly the concepts beforehand, how did programmers even come up with the most efficient software?

Sure, computers can do many things, but to me, it is still wise not to be overly reliant on them, which is actually what Uncertain and I meant. :D

Ah ha I was waiting for someone to say... "if ur GC break down in the exams...". The point is... are you learning for the sake of knowledge or for the sake of examinations? :)

And let's say it really break down... do you think you have time to find turning points, asymptotes roots etc etc manually in an exam which timing allocations is geared toward use of electronic aids even if you are reasonably proficient at it?

2nd question is invalid as the concepts are still properly taught in A Maths beforehand except that you do not go in depth. They still test asymptotes plus behaviour as curve approach asymoptotes and infinity, turning points, roots and etc in GCE O level A Maths. Just that they are used in basic scenarios and not explored at large. Please do not forget that the O and A are GENERAL certifications of education... not specialised certifications. They should only need to provide a foundation which will equip a student to have a broad superfical understanding of the subject matter. Should a student choose to go further in specialisations... what do you think degree courses are for? Of course we expect maths degree holders to program, not A level students. In society, the majority only need a broad understanding of subjects while a small minority specialises. Your friendly housewife need only to know how to cook a variety of homecook dishes... and when you want to eat Man Han full banquet... you go to the chefs.

Ok cool... so it is no good to rely on the chefs to dish up Man Han full banquet and every housewives should know how to prepare these eh?

purpledragon84 SG

wah.. i neber use GC before.. scientific calculator nia.. and poisson distribution can do with scientific calculator wat.. press until high..

Uncertain SG

Ah ha I was waiting for someone to say... "if ur GC break down in the exams...". The point is... are you learning for the sake of knowledge or for the sake of examinations? :)

Ok cool... so it is no good to rely on the chefs to dish up Man Han full banquet and every housewives should know how to prepare these eh?

Well Mikethm, I cannot change your thinking. But if u ever want to pursue study in university especially in the finance sector or mathematices field, then good luck to u.

As what purpledragon had said, it is possible to do all math questions with scientific calculator. The only time i use my GC is for my matrix reduced-row echleon form and my Further Mathematics teacher taught us this topic by first telling us to do it the brute force way. I admit it is tough but is enriching which make some of my courses a brisk.

tinuviel07 SG

wait for someone to get stuck and goes back to square 1 then they will know the importance

eagle SG

Ah ha I was waiting for someone to say... "if ur GC break down in the exams...". The point is... are you learning for the sake of knowledge or for the sake of examinations? :)

Ok cool... so it is no good to rely on the chefs to dish up Man Han full banquet and every housewives should know how to prepare these eh?

The point is... are you learning for the sake of knowledge or for the sake of examinations?

Ah, simple

Do you learn how to use the GC only for the sake of knowledge, or do you learn how to apply the concepts without relying on the GC for the sake of knowledge? I still believe in the latter

Err.... I'm talking about A level H2 Maths and the use of graphics calculators.

Again, if you are trying to say that it is because O levels and A levels are "GENERAL certifications of education", then you are not learning it for the sake of knowledge, but for the sake of examinations. This clearly contradicts what you said earlier

Because it is for the sake of knowledge, I always teach extras to my students (if they want to hear) and tell them what they are learning at their level is not complete, to help them appreciate the subject at hand. I believe that one should adopt the same mindset for maths as well, learn and not rely on the GC.

Of course, I'm not denying that you will still have to be well trained in the GC as well to make full use of it in exams; I strongly advocate that too.

Mikethm SG

Yes Yes I know you are talking about H2 maths.The point is that THEY CHANGED the O level A Maths to fit in with the progression to H2 Maths. They are linked.

Saying that the O and A are general certification does NOT mean you are learning for the paper chase. It simply mean the focus of the syllabus is to equip the student with the foundation in whatever subject examined. As to the attitude of the student, it depend on the individual. Eg. A student learning complicated sketching techniques so that he can sketch for the exams if the GC break down... is learning for the sake of examinations. A student satisfied with concepts of graphs and more than happy to let the GC/Computer do the actual mechnical aspects is learning for the sake of knowledge. Why? Because it is a fact that in real life, one can rely on computer to sketch BUT need the concept base knowledge to understand the output.

And if we disagree on the importance of the need for manual complicated graph sketching, just let it lie. Opinions are opinions. And the only one that matters is your own. Because in your mind, you are always right. Thus no point arguing about this.

eagle SG

Erm

The problem is, the graph sketching techniques are not complicated at all. We are not even talking about the super chim graphs, just standard graphs at A levels. It is definitely not at all tough to sketch them.

Next, "A student satisfied with concepts of graphs and more than happy to let the GC/Computer do the actual mechnical aspects is learning for the sake of knowledge."

This is what I am saying actually. However, is the student even taught the concepts of the graphs at all to let them be satisfied? It doesn't seem so. Do you think most of the A level students now have the ability to get a general feel of a graph before even touching the GC? Do you think they really understand why certain parts of a graph are of an increasing or decreasing nature? And the short cuts to knowing whether a graph tends to positive or negative infinity when it encounters an asymptote? Even if you give them the asymptotes, the stationary points, the x and y intercepts (and thus not necessary to do all the so-called complicated stuff), they might not even be able to sketch a nice and proper graph with ease. These are to me also concepts of the curve; merely understanding the output from the GC is insufficient.

These are all simple concepts, but sadly, with the over-reliance on the GC, many of the students are now not able to employ them. Not sure if you considered these "in-depth". But from what you have said, I think we are sort of going a little off tangent to each other, that's why the disagreement

Uncertain SG

Yes Yes I know you are talking about H2 maths.The point is that THEY CHANGED the O level A Maths to fit in with the progression to H2 Maths. They are linked.

If u think ur GC can draw graph, i challenge u to use your GC to draw the graph (ax+b) / (cx +d) where a, b,c, d are constants . Tell me all the asymptotes from your GC.

If not fair, use ur GC again draw the graph 4(x+4)^2 - (y-2)^2 = 1. Tell me the asymptotes.

I want to see how ur mighty GC gives me all these answers.

secretliker SG

Well of course there are still questions in the 'A' Level Syllabus which requires manual sketching. The GC is there to help one gauge the general shape of an equation, however one must know how to find intercepts and asymptotes as my current GC do not directly tell me.

The more challenging types of questions are those which they give you symbols instead of actual numbers, where you are required to find certain points in terms of symbols like y-intercept = a/b or something like that. But even this should be basic concept.

Simply put, the GC is there as an aid, a helpful aid to simplify tedious calculations, which would otherwise waste precious time, and not part of the testing objective essentially.

For instance, a question which involves solving for the roots of say an equation y = x^4 - x^3 + x^2 - 1, but the main purpose of that question is not to test students on solving for the roots, but on other stuff. Thus the process can be simplified with the GC and just read off the roots from the GC.

In another anology, most 'A' Level students, I believe, are familiar with the rules of differentiation and integration, but they do not know how to derive the exact principle of why d/dx (sin x) = cos x, from scratch. Hence, for simplicity sake, we take for granted that the equation above is true and apply it whenever required in examinations, because what the Examiner intend to test was not for one to show why d/dx (sin x) = cos x, but for one to show the working using simplified and already-worked out formulae.

eagle SG

The GC is there to help one gauge the general shape of an equation

Which means you still need the GC to gauge the general shape of an equation even though you might already be given the intercepts, stationary points, asymptotes and gradient information?

In another anology, most 'A' Level students, I believe, are familiar with the rules of differentiation and integration, but they do not know how to derive the exact principle of why d/dx (sin x) = cos x, from scratch. Hence, for simplicity sake, we take for granted that the equation above is true and apply it whenever required in examinations, because what the Examiner intend to test was not for one to show why d/dx (sin x) = cos x, but for one to show the working using simplified and already-worked out formulae.

Erm... the bolded one is really basic A level trigo combined with understanding how the first principles of differentiation works

Simply put, the GC is there as an aid, a helpful aid to simplify tedious calculations, which would otherwise waste precious time, and not part of the testing objective essentially.

Yep. What Uncertain and I meant is that over-reliance is not good, but one should make full use of it when possible during exams to simplify the tedious calculations.

Analogy, you know how to do big numbers multiplication from primary school. You understand the principles of how to solve it, but you use a normal calculator as an aid to simplify it. But then, you know you can still do it even if your calculator goes haywired.