We have that . This is what we call the recurrence formula. We consider this to give us the ratio between successive probabilities. And to illustrate how this works, nothing beats an example question.

Consider candies are packed in packets of 20. On average the proportion of candies that are blue-colored is . It is know that the most common number of blue-colored candies in a packet is 6. Use this information to find exactly the range of values that can take.

First, most common number is the same as saying the modal/ highest frequency.

This means that is the highest/ largest probability… Let us turn our attention to the recurrence formula now. If is the largest, then it means that and also .

Lets start by looking at the first one…

But hold on! This looks like the recurrence formula. (ok, in exams, its either you use the recurrence formula or derive on the spot. Both works!)

I want to share a question that is really old (older than me, actually). It is from the June 1972 paper. As most students know, Maclaurin’s series was tested in last year A’levels Paper 1 as a sum to infinity. And this DRV did the same thing. Here is the question.

A bag contains 6 black balls and 2 white balls. Balls are drawn at random one at a time from the bag without replacement, and a white ball is drawn for the first time at R th draw.

(i) Tabulate the probability distribution for R.

(ii) Show that E( R ) = 3, and find Var( R ).

If each ball is replaced before another is drawn, show that in this case E( R ) = 4.

I was teaching scatter diagram to some of my students the other day. A few of them are a bit confused with correlation and causation. I gave them the typical ice cream and murder rates example, which I shared here when I discussed about the r-value.

Think of correlation like a trend, it simply can be upwards, downwards or no trend. And since we only discuss about LINEAR correlation here, strong and weak simply is with respect to how linear it is, that means how close your scatter points can be close to a line.

Since A’levels, do ask students to draw certain scatter during exams to illustrate correlation. Here is a handy guide.

Here is the suggested solutions for H2 Mathematics (9758). They are all typed in LaTeX, so if it does not render, please leave a comment and let me know. Thank you.

The suggested solutions for H2 Mathematics (9740) is here.

Students of mine should obtain the modified A’levels Paper, and the solutions to the additional questions can be found here.

This show that there is a strong positive linear correlation between the weekly earnings of an employee and the number of years the employee is with the company.

(iii)
Using GC,

(iv)

Let

Thus, the employee earns an estimated weekly earnings of $293.

(v)
Firstly, the linear model is not appropriate here. Since it suggests as the number of years increases, the weekly earnings will increase proportionately, which is not realistic.

Secondly, is out of data range and this is extrapolation, which is a bad practice since our trend might not continue out of data range.

There is a probability of 0.0366 of observing a result equal to or more extreme than that is actually observed when the mean volume is indeed 0.6.

(ii)
It is not necessary since the sample size of 50 is sufficiently large for us to approximate the volume of juices to a normal distribution by Central Limit Theorem.

*For students: We half our values here as we scale by factor .

x- intercept
y- intercept
(ii)

*For students: We plus one to our values here as we translate by 1 unit in the positive x direction.

x- intercept
y- intercept inconclusive

(iii)

*For students: We plus one to our values here as we translate by 1 unit in the positive x direction. Then, we half our values here as we scale by factor .

(i)
The average number of customers joining a supermarket checkout queue is a constant.
A customer joining a supermarket checkout is independent of another customer joining a supermarket checkout.

(ii)
Let denote the number of customers joining a supermarket checkout queue in a 5-minute period.

(iii)

(iv)
Let denote the number of customers leaving a supermarket checkout queue in a 3-minute period.

(ii)
Let X denote the mass of biscuit bars, in grams
Let denote the population mean mass of biscuit bars, in grams.

Test
against at 1% level of significance

Under approximately, by Central Limit Theorem.

Test Statistic,

Using GC,

Since , we do not reject at 1% level of significance and conclude with insufficient evidence that the mean mass differs from 32g.

(iv)
The sample size, is sufficiently large for the manager to approximate the the population distribution of the masses of the biscuit bars to a normal distribution by Central Limit Theorem.

(iii)
is within given data range, and we are performing interpolation, which is a good practice.
The r value is close to 1 which suggest a strong positive linear correlation between the average yields of corn and the amount of fertiliser applied.

(i)
The probability that the lights are faulty is constant.
The event that the light is faulty is independent of another light being faulty.
The light can only be either faulty or non faulty.

(ii)
Let denote the number of faulty lights in a box of 12.

(iii)
Required probability

(iv)
Let denote the number of faulty lights in a carton of 240.

Since and ,

approximately.

by continuity correction

(v)
Events in (iii) is a subset of events in (iv).

(vi)

(vii)
Required Probability

(viii)
From (vii), the quick test seem to be 94% accurate. However, from (vi), we understand that out of the number of lights identified faulty, 42.1% of them will be a mistake. As such, the quick test is not worthwhile, since light identified as faulty are mistakingly discarded. Moreover it will cost money to administer a test.

Students of mine who have been diligently doing the modified TYS I sent them, and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, these are modified to cater to the 9758 syllabus…

The rest of the solutions (that are questions from the original TYS) can be found here.

*For students: We half our values here as we scale by factor .

x- intercept
y- intercept
(ii)

*For students: We plus one to our values here as we translate by 1 unit in the positive x direction.

x- intercept
y- intercept inconclusive

(iii)

*For students: We plus one to our values here as we translate by 1 unit in the positive x direction. Then, we half our values here as we scale by factor .

(i)
Each biscuit in the list of all biscuits produced has equal chance of being selected independently of one another.

(ii)
unbiased estimate of population mean

unbiased estimate of population variance

(iii)
Let X denote the mass of biscuit bars, in grams
Let denote the population mean mass of biscuit bars, in grams.

Test
against at 1% level of significance

Under ,since is sufficiently large, approximately, by Central Limit Theorem.

Test Statistic,

Using GC,

Since , we do not reject at 1% level of significance and conclude with insufficient evidence that the mean mass differs from 32g.

(iv)
The sample size, is sufficiently large for the manager to approximate the the population distribution of the mean masses of the biscuit bars to a normal distribution by Central Limit Theorem.

(iii)
is within given data range, and we are performing interpolation, which is a good practice.
The r value is close to 1 which suggest a strong positive linear correlation between the average yields of corn and the amount of fertiliser applied.

(i)
The probability that the kitchen lights are faulty is constant.
The event that the kitchen light being faulty is independent of another kitchen light being faulty.

(ii)
Let denote the number of faulty lights in a box of 12.

(iii)
Required probability

(iv)
Let denote the number of faulty lights in a carton of 240.

(v)
Events in (iii) is a subset of events in (iv).

(vi)

(vii)
Required Probability

(viii)
From (vii), the quick test seem to be 94% accurate. However, from (vi), we understand that out of the number of lights identified faulty, 42.1% of them will be a mistake. As such, the quick test is not worthwhile, since light identified as faulty are mistakingly discarded. Moreover it will cost money to administer a test.

So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. ðŸ™‚