# Probability assistance required

• Jan 31st 2011, 01:56 AM
Glitch
Probability assistance required
The question:

Down's syndrome is a disorder that affects 1 in 270 babies born to mothers aged 35 or over. A new blood test for the condition has a sensitivity (i.e. the probability of a positive test result given the Down's syndrome is present) of 89%, The specificity (i.e. the probability of a negative test result given that Down's syndrome is absent) of the new test is 75%.

a) What proportion of women over the age 35 would test positive on this new blood test?
b) A mother over age 35 receives a positive test result. What is the chance that Down's syndrome is actually present?
c) A mother over age 35 receives a negative test result. What is the chance that Down's syndrome is actually present.

My attempt:
For a) I managed to get the answer via a tree diagram, which is 25.237% However, I'm not sure how to attempt b) and c). How should I attempt this? Thanks
• Jan 31st 2011, 05:11 AM
tom@ballooncalculus
Just in case a picture helps...

If you can see how your tree diagram relates to this one...

http://www.ballooncalculus.org/draw/venn/two.png

(with P = testing positive for the Down's, N = testing negative, D = Down's present, A = Down's absent)

... then you might be able to see how I've rearranged it again here...

http://www.ballooncalculus.org/draw/venn/twoa.png

... and how to use this.

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• Jan 31st 2011, 06:23 PM
Glitch
Hmm, my tree is different. I made each branch a fraction:

http://i.imgur.com/RIVWc.jpg

Yours is confusing me. :/
• Jan 31st 2011, 09:21 PM
Glitch
I ended up working it out using Bayes Rule. I'd still like to understand how to do it via a tree diagram though.
• Jan 31st 2011, 11:48 PM
tom@ballooncalculus
Quote:

Originally Posted by Glitch
I ended up working it out using Bayes Rule. I'd still like to understand how to do it via a tree diagram though.

Right! The tree method is a good way of understanding Bayes.

Compare yours with my first. I used 89% and 11% like you, but imagined that was 89 mothers and 11 mothers, making 100 in all. Then 269 times as many as these is 26900, and we share these 25:75.

(I.e. replace probabilities with possible quantities, as recommended in passing but curiously not exemplified here Conditional probability - Wikipedia, the free encyclopedia under 'The Conditional Probability Fallacy').

This way we get a venn-diagram sense of the probabilities. The second tree is I think slightly better than rearranging the venn-diagram itself as here http://www.mathhelpforum.com/math-he...ty-169248.html. (Also http://www.mathhelpforum.com/math-he...on-169053.html).
• Feb 1st 2011, 12:20 AM
Glitch
Ahh I see. So to get the solution to b), it's just a matter of 89/6814. I should try a few more questions. Thanks.