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My close friend found out he had cancer and then received treatments.

Based on medical studies (I found in medical journals), I was able to find out that only 5% of the patients with this one condition/test scores (A) experienced recurrence. I was also able to find out results from a different study that looked at a different condition (B). These results showed that recurrences was experienced in 10% of these patients with condition B (these results are from a separate medical studies that just reported on recurrence rates of patients from the 2 separate tests/conditions).

These 2 conditions probably have very high levels of correlation, but can be exclusive. You can have one without the other (working both ways).

Given that he has both conditions (Group A with 5% AND Group B with 10%), what is the overall probability of the cancer coming back based just on these 2 variables? Is it (0.05 X 0.10) = 0.005? or is it something else?

I understand there are a lot of other variables involved, but from a statistical point of view based on just these 2 probabilities that i know of... how would i calculate the overall probability of the cancer coming back?

thank you.

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    $\begingroup$ My deepest sympathies to you and your friend, but I hardly think we can give you any sort of help here that a doctor couldn't do a better job of providing. I hope all works out for you in the end. $\endgroup$ – The Count May 27 '17 at 17:32
  • $\begingroup$ Is this a hypothetical question? If not, you should ask a doctor, there's far too many variables and far too little information here for anybody to comment on it. If it is hypothetical, then we can assume that the cancer has some chance of recurring due to A, and some due to B. Then, the total probability of recurring is 1 minus the probabilty of not recurring, i.e, $P = 1-(1-0.05)(1-0.1)$ This assumes that A and B are independent though, so no correlation. $\endgroup$ – Guy May 27 '17 at 17:33
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In general, events $A$ and $B$ are independent if (and only if) $P(A \cap B) = P(A)P(B)$. It is unlikely that this is true in your case so the best you can say is $P(A \cap B) \le P(A) = 0.05$. This is based on a frequentist philosophy (i.e. it gives information only about a large group of people, not a single case). If you want information about your friend's case that needs to come from a healthcare professional that knows your friend's situation.

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