A tree might help.
Has the disease. This is not disputed.
Correct Diagnosis 60% -- No Correct Diagnosis 40%
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Now, consider just the Correct Diagnosis: (60%)
Death any way: 40% (I presume you don't actually mean that the treatment itself killed the patient.) -- Survival: 60%
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Now, consider just the Missed Diagnosis: (40%)
Death: 70% -- Survival: 30%
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So, what do we know and how can we answer the question? Bayes might help.