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[Nobbylad]

But can you lay bricks, sweat/mortice a joint, dig a trench, flash a chimney, skim a wall, find water or make bread?

1st world problems #shrug

@Nutsinatin, how did you do those calculations without a calculator? (This isn't a criticism, I need to know!)

Originally Posted by allantheboss

Question 1:
Approximately 1 in 14 men over the age of 50 has prostate cancer. The level of 'prostate specific
antigen' (PSA) is used as a preliminary screening test for prostate cancer.
7% of men with prostate cancer do not have a high level of PSA. These results are known as 'false
negatives'.
75% of those men with a high level of PSA do not have cancer. These results are known as 'false
positives'.
If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?

I have read this over many times now and I disagree with your answers to Question 1.

The main problem with your reasoning is that the two factors i.e. having prostate cancer and high PSA are linked. This means that as soon as you know something about the result of the PSA test then the "1/14" chance of having cancer is no longer valid as this represents the population as a whole . Your 0.5% is the probablility of someone having normal PSA AND prostate cancer in general, not the probability of someone having cancer IF they have normal PSA.

I think that the only way of finding the answer is to fully delineate the two pieces of information.

Here we need to read the question extremely carefully.... mainly because it is written badly (see below)!!!

Breaking the question down, every man over 50 belongs to one of four groups. We know that 1 in 14 have prostate cancer (groups A&B) and that of those who have prostate cancer, 7% do not have high PSA. Now all we know about the groups who do not have prostate cancer is that they are the other 13 in 14 but we can safely assume that a fraction "x" of them also have high PSA. Laying that down we get:

A) Prostate cancer & high PSA
Fraction(A) = 1/14 * 93% = 93/1400

B) Prostate cancer & normal PSA
Fraction(B) = 1/14 * 7% = 7/1400

C) No prostate cancer & high PSA
Fraction(C) = 13/14 * x

D) No prostate cancer & normal PSA
Fraction(D) = 13/14 * (1-x)

What the question asks is "If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?". If you have normal PSA then you belong to groups B) or D) and if you have cancer then you belong to group B) so we are looking for the ratio between Fraction(B) and [ Fraction(B) + Fraction(D) ]. However, we have a problem --> we don't know D).

Now... I think the wording of the question is VERY tricky...for example, by switching the word order around for the 7% vs 75% they have altered the meaning of the two phrases.

"75% of those men with a high level of PSA do not have cancer." So: all men with high PSA represents Fraction(A) + Fraction(C) and of those groups, 75% do not have cancer i.e. Fraction (C).

So if it is 75% Fraction (C) and 25% Fraction(A) then that means that
Fraction(C) = three times more than Fraction(A)
Fraction(C) = 3 * Fraction(A)
Fraction(C) = 3 * 93/1400
Fraction(C) = 279/1400

We can now simply find D) using fact the probability of all options summed must always equal 1:
Fraction(D) = 1 - Fraction(A) - Fraction(B) - Fraction(C)
Fraction(D) = 1400/1400 - 93/1400 - 7/1400 - 279/1400
Fraction(D) = 1021/1400

Again, normal PSA means categories B) & D) and the probability of having prostate cancer given a normal PSA is the ratio of Fraction(B) to [ Fraction(B) + Fraction(D) ].


P(normalPSA --> cancer) = Fraction(B) / [ Fraction(B) + Fraction(D) ]
From above:
P = (7/1400) / (7/1400 + 1021/1400)
Combine the second fraction:
P = (7/1400) / (1028/1400)
Cancel out the 1400s and we get:
P = 7/1028
P = ~1%


This is my "final answer"!





But..........I deliberately omitted the second half of the phrase "75% of those men with a high level of PSA do not have cancer. These results are known as 'false positives'.", because 'false positives' actually describes group C) which is inconsistent with the first half of the sentence. However, if you take it that 75% are "false positives" i.e.
Fraction(C) = 0.75 = 1050/1400
Then:
Fraction(D) = 1 - Fraction(A) - Fraction(B) - Fraction(C)
Fraction(D) = 1400/1400 - 93/1400 - 7/1400 - 1050/1400
Fraction(D) = 250/1400
Again:
P(normalPSA --> cancer) = Fraction(B) / [Fraction(B) + Fraction(D) ]
P = (7/1400) / (7/1400 + 250/1400)
P = (7/1400) / (257/1400)
P = 7/257
P = ~3%

Still the test comes out pretty well on the surface.... but mainly because the chances of having prostate cancer in the first place is low. Interpreted in this way, there is more chance of the test being wrong than right if you do not have cancer!

[Nutsinatin]

Originally Posted by allantheboss

@Nutsinatin, how did you do those calculations without a calculator? (This isn't a criticism, I need to know!)

Practise as I'm working in a lab whilst doing my chemistry degree part time!
But in seriousness they gave you some nice numbers that give you whole numbers as answers.

I like to keep this in standard form as it stops you having to worry about decimal places or long division, but simplified this in my answer as I couldn't be bothered to type it all last night when I was tired. So you take your 0.420 and turn it into 42x10^-2. This makes it nice and easy to divide by 84, so 42(x10^-2) gives you 0.5(x10^-2) then when you neaten that up you get 5x10^-3.

For the next bit you need to be aware of some of the laws of indices, namely if you multiply powers you add them, or if you divide them you subtract them. For example 10^5 x 10^6 is the same as 10x10^(5+6) = 100^11. Just remember to multiply/divide the bases as well!

So to multiply the number of moles by the Avogadro constant 6x10^23 do 5(x10^-3) x 6(x10^23) = 5x6(x10^-3+23) = 30x10^20. Then you neaten this up to 3x10^21.

The last calculation is also nice and easy as you have 3(x10^21) x 12, which is 3x12(x10^21) = 36x10^21, which is 3.6x10^22 when you neaten it up.

Hopefully that helps but let me know if there are any bits I skipped over or if there's anything you would like me to clarify! When is your test then?

Originally Posted by Nutsinatin

Practise as I'm working in a lab whilst doing my chemistry degree part time!
But in seriousness they gave you some nice numbers that give you whole numbers as answers.

I like to keep this in standard form as it stops you having to worry about decimal places or long division, but simplified this in my answer as I couldn't be bothered to type it all last night when I was tired. So you take your 0.420 and turn it into 42x10^-2. This makes it nice and easy to divide by 84, so 42(x10^-2) gives you 0.5(x10^-2) then when you neaten that up you get 5x10^-3.

For the next bit you need to be aware of some of the laws of indices, namely if you multiply powers you add them, or if you divide them you subtract them. For example 10^5 x 10^6 is the same as 10x10^(5+6) = 100^11. Just remember to multiply/divide the bases as well!

So to multiply the number of moles by the Avogadro constant 6x10^23 do 5(x10^-3) x 6(x10^23) = 5x6(x10^-3+23) = 30x10^20. Then you neaten this up to 3x10^21.

The last calculation is also nice and easy as you have 3(x10^21) x 12, which is 3x12(x10^21) = 36x10^21, which is 3.6x10^22 when you neaten it up.

Hopefully that helps but let me know if there are any bits I skipped over or if there's anything you would like me to clarify! When is your test then?

Amazing explanation! Thank you so so much! Are you at Surrey Uni? I'm at Holloway.

[Nutsinatin]

No I'm at Manchester Metropolitan, and do it mostly through text books, online modules, webcam and online tests, with a few paper tests invigilated by my manager at work as they're funding my degree. Most of my practical work I do at work and send my results and photos etc. as a report, and I go up to the uni for a week each year to do some more practicals starting at 8 in the morning and finishing at 10 at night.

[Nutsinatin]

Originally Posted by Spank86

That's a lot more in depth than the gcse science I did.

Closer to some of the A level questions.

I didn't cover any of the chemistry at GCSE (five years ago) as my school chose double science which is triple science will all the hard units removed (as we didn't have any science teachers who had studied any science discipline) but it's definitely on the syllabus now for anyone who takes triple science, especially with nice numbers like those. At A level they like to make the numbers much more awkward and get you to manipulate this sort of data a lot more.

Originally Posted by a1istair

I have read this over many times now and I disagree with your answers to Question 1.

Some interesting points you put forward, I will mull over this one and get back to when I haven't got a Thursday deadline for my own uni work which I should really be getting on with now!

I can wipe my own ass.

Originally Posted by Owenski

I can wipe my own ass.

Now you're just showing off Matt.

Well I felt left out, Ive been away for a little bit. Come back to pop in and this place turns to Mensa in my absence.