# Plot of Probability of Cancer vs Test Sensitivity

I'm looking at the following problem and its solution and am really struggling to understand how they came up with the necessary values.

My Questions

1, For part (a) we are asked to plot $$y=P(C|+)$$ as a function of $$x=P(+|C)$$ given that P(C)=0.01. The formula (3.87) can then be translated to $$y=\frac{0.01x}{0.01x + 0.99P(+|NC)}$$. I don't know what they use for $$P(+|NC)$$, nor how it could produce the blue straight line graph that they claim in the solution? I have a similar question about how changing 0.01 to 0.3 will be able to produce the orange curve since it doesn't appear to change the $$x$$ dependence?

2, Similar problem to above for the second graph - I don't see how we have all the values? Here we use $$x=P(+|NC)$$ but we don't have $$P(+|C)$$?

3, The claim is that the third graph uses $$y=P(C|+)$$ and $$x=P(C)$$ using $$P(+|C)=0.9$$ and $$P(+|NC)=1-0.92=0.08$$. Plotting it on WolframAlpha I get something similar so I can believe this one. However, I would like some explanation of the third paragraph on page 13 which says the biggest information gain is when $$P(C) \sim 0.3$$. Presumably this is because there is the biggest "gap" between blue and black curves? And since the black curve corresponds to assigning $$P(C|+)=P(C)$$ it is basically a useless test that adds no info, right?

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The Original Question and Solution

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• can you share the source? and there is a typo in (1), $1-x$ should be $0.99$ – gunes Feb 26 at 14:25
• @gunes fixed typo and see page 13 on here google.com/url?sa=t&source=web&rct=j&url=http://… – user11128 Feb 26 at 14:47
• @gunes presumably we need to guess a value for P(+|NC) in terms of x? I can't see how we would work it out using e.g. a Venn diagram... – user11128 Feb 27 at 9:39

It appears that you missed the general information given in the sub-section beginning, i.e. sub-section $$3.8$$ (Page 12). That paragraph makes a general statement valid for all questions under section $$3.8$$ unless stated otherwise. So, you'll assume FPR = 0.08, which is the missing information in part (a), i.e. $$P(+|NC)$$. I've used $$P(+|NC)=0.08$$, plotted your equation, and obtained same graph given in the solution:
Again, for (2), you have $$P(+|C)$$ value given as $$0.9$$. For (3), your comment is correct. It has the biggest gap around $$P(C)\approx 0.3$$ and this gap represents the information you've gained by applying the test vs no-test.