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You have torn a tendon and are facing surgery to repair it. The surgeon explains the risks to you: infection occurs in 8% of such operations, the repair fails in 17%, and both infection and failure occur together in 2%. What percent of these operations succeed and are free from infection?

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Answer:

77% of the operations succeed and are free from infection.

Step-by-step explanation:

Consider the following events. A = An infection occurs, B = the repair fails. Then, we are given the following probabilities P(A) = 8%, P(B) = 17% and P(A[tex]\cap[/tex]B) = 2%. We are asked for the following probability [tex]P(A^c \cap B^c)[/tex]. First, recall the following formulas.

[tex]P(A^c) = 1- P(A)[/tex]

[tex]P(A\cup B) = P(A)+P(B)-P(A\cap B) [/tex]

[tex](A\cup B)^c = A^c \cap B^c[/tex] (known as Demorgan's Law)

Note that we are asked to calculate [tex]P(A^c \cap B^c)[/tex] which is equivalent to [tex]P((A \cup B)^c) = 1 - P(A\cup B)[/tex]. Using the formulas, we have that [tex]P(A\cup B)[/tex] = 8%+17%-2% = 23%. Then, [tex]P(A^c \cap B^c)= P((A \cup B)^c) = 1 - P(A\cup B)[/tex] = 1-23% = 77%

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