The nutrition and food science division of Farnsworth Pharmaceuticals is alpha testing an OTC diagnostic kit so that adults who self-report a certain food allergy can affirm their insensitivity to that food allergen. Preliminary clinical trials yielded that 25% of the volunteers who suspected they had food allergies were clinically identified as sensitive or intolerant to the food allergen; unfortunately and disappointingly, only 10% of those who tested positive in the clinic also tested positive with the kit. Furthermore, of those self-reporting adult volunteers who had a negative clinic outcome, 5% tested positive anyway with the kit.

Using Bayes Theorem, what is the chance of a true positive outcome of this compound event: BOTH a positive result on the clinical diagnosis AND a positive result on the OTC kit diagnosis, given that the clinical diagnosis was positive? HINT: What is the appropriate arithmetic operation to compute the probability of a compound event that utilizes the both ... and ... conjunction? Express the answer as a percentage without insignificant digits, with percent symbol.

Can you show using the tree diagram? I'm having the hardest time knowing how to use it here.