1:2:1 Maths Tuition

Another - What Did He Get Next?

Query: Rob’s marks in 5 tests were 60%, 55%, 45%, 40% and 50%. What mark must he get in the last test to gain an overall mean of 55%?

Response:

Gender Balancing Act.....

Query: A committee consists of 6 men and 4 women. A sub-committee of 4 members is to be formed at random. Find the probability that the sub-committee contains at least 1 member of each sex?

Response:

Where's The Median?

Query: How do I calculate the median of a continuous random variable, when all I'm given is the p.d.f.?

Response: Take a look at a worked example on this topic, by clicking the graphic below...

 

Continuous Random Variable...

Query: A continuous r.v. has probability density function 

f (x) = k(1 - x) for 0<x<1 and f (x) = 0 otherwise.

Evaluate k.

Response: 

Click on the graphic below...

How Do I Find The Mean Of This Data?

Query: I've got data for the number of children absent from a certain class. The data is already in a table [supplied]. How do I find the mean of this data?

Response: Please click the graphic to view the data and the calculations to arrive at the mean. 

Probability Density Function...

Query: A random variable has a pdf of f(x)=k(1-x)², for x between 0 & 2 and f(x)=0 elsewhere. How do I find the value of 'k'?

Response: The key point here is that the integral of f(x) between the limits of 0 and 2 must be 1, as it is a p.d.f.

Please click on the graphic for the solution.

What's A Causal Link?

Query:  Explain, briefly, with the aid of an example, what is meant by a ‘spurious’ or ‘nonsense’ statistical correlation.

Response: High values of the correlation coefficient can sometimes be found between two variables for no apparent reason. This is a particular danger when vast amounts of multi-variate data are mined and analysed.

Sometimes there is a causal link between each variable producing the correlation and a third variable which triggers the large coefficient.

A simple example can be found in crime statistics of a local authority. If you run an analysis between the variables (1) the number of police arrests for street drunkeness (2) the number of places of worship within one mile of the arrest, you'll probably find a significant correlation, but where's the likely causal link?

Each of these variables is likely to be positively correlated with the population of the area in which they occur. High population areas are often a focus for drunkeness as there are a large number of bars and clubs to service the demand and will also have a higher density of places of worship, which would probably account for the 'spurious' correlation found.

I remember reading a paper by a researcher from Cardiff University in which he analysed a large amount of quantitative schools' results and student personal characteristics data and found a significant statistical correlation between the Mathematics results of individual schools and the altitude (i.e. the number of metres above sea level) of the school buildings.

His tongue-in-cheek conclusion was that, if possible, schools in low altitude areas should adopt the strategy of improving their student scores at Mathematics by ensuring that all lessons took place upstairs!

Any Chance Of A Raffle?

Query: Ken and Hilda have each bought two tickets in a raffle for which one hundred tickets have been sold. There is only a first prize and a second prize. What is the probability that they don't win anything between them?

Response: Please click on the graphic below to see the solution.

Eyes Down...

Query: Twenty one identical balls are numbered 1 to 21. They are placed in a bag and two balls are selected without replacement. What is the probability that the numbers on both balls are less than 7?

Response: Click on the graphic for the solution...

Spin Out Of Control...

Query [from LukeP]:

The spinner gives the following output data: w,b,b,w,r,r,g,b,w,b,r,g,g,r,w,w,r,g,w,b.

I have to find the median colour of the 20 spins? 

Response: I fear there is no median here, as the data is not numerical. Please go back to the source of the question and raise a query.

Remember: The median is the middle value of ordered data and it is only possible to order numbers.

It Doesn't Fit The Table!

Query [from KN]: A variable X has a Binomial Distribution, where n=25 and p=0.8. I'm meant to work out P(X>15), but my Binomial Table doesn't list p-values above 0.5?

Response: As this is a Binomial Distribution, while you are counting 'success' you are counting 'failure' also at the same time.

So, think of Y~B(25,0.2) instead and look up your tables for P(Y<10).

An Odds On Certainty?

I would never have thought that going to the races would be considered an useful mathematical exercise.

Well, if you follow this link to the recent video story which appeared in the Education Guardian (UK) website you will find out that someone obviously does!

I presume that actually betting on the races is optional and on a positive note, the children might begin to realise the true probability of picking a winner, with any luck.

A Chance Lesson...

We're grateful to DWS for drawing our attention to the following video lesson, posted on YouTube, about the basics of probability by keithpeterb.

It's well worth a viewing, especially if you're looking for a quick revision of the basics of this topic, but you'll have to ignore the references to exercises and units!

Here We Go: A, B, C,....

Query: This fair spinner gives the following output:

A, B, A, C, A, C, D, A, A, C, A, B.

Write down the probability of getting an A and what do you expect to happen to this probability as the number of spins increases?

Response: The letter A has occurred 6 out of the 12 spins, so p(A)=6/12=0.5

As the number of spins increases, there should be a strong tendency for p(A) to reach 0.25, if the spinner is as fair as you say.

Keeping Afloat...

Submitted Question [from TracyJ]: The probability of a swimmer winning a race is 84% and he swims 50 races in a season. How many races would you expect him to win?

Repsonse: By multiplying the probability of an event happening by the number of 'identical' events, you arrive at the expected number of successes.

So, your answer of 42 wins was actually correct (i.e.50x0.84), but I can understand how you felt it couldn't be that easy!

1:2:1 Challenge: (Solution 07/01)...

Although there are 10 letters in the word STATISTICS, some are repeated.

The letter count is: THREE Ss & Ts, TWO Is, ONE A & C.

This means that the number of different arrangements is 50,400!

Download Solution [pdf]

There were two correct solutions, submitted by KK2 (with solution) and LP (answer only). Well done!

The Plot Thickens...

Query [from DJ99]: This data represents the lengths in cm. of something or other. I've been asked to create a Box and Whisker diagram for the data? Help!

Response: In order to proceed, it is necessary to evaluate the following statistics from the dataset you supplied:

• Minimum data item  (whisker bottom)
• Maximum data item (whisker top)
• Lower Quartile (box bottom)
• Median (in the box, but not always exact middle)
• Upper Quartile (box top)

These values are then plotted to scale, as shown in the diagram.

1:2:1 Challenge (Problem: 07/01)...

See if you can answer the following problem:

Find the number of arrangements of the letters of the word STATISTICS.

Please e-mail us your answer and explanation - the solution and all comments received will be posted at 12 noon [GMT] on 01/09/07.

There are no prizes, sorry - just the challenge - but are you up to it?

What is Normal?

Query [from RDE]: We're given that Y is Normally distributed with a mean of 56 and a standard deviation of 10. I calculated the probability of Y being above the mean but less than 65 as 0.1841, but this is wrong?

Response: Here, we want to find p ( 56 < Y < 65 ), which is represented by the shaded region in the diagram below...

Download Solution [pdf]

Venn Rules

Query [from RDE]: I’ve got confused about overlapping sets within a probability space, using Venn Diagrams?

Response: The problem you supplied had the following data:

leading to the Venn Diagram...

Download Full Question & Solution [pdf]