Question

Suppose John has a torn tendon and is facing surgery to repair it. The surgeon explains the risks to John: infection occurs in 4 % of operations, the repair fails in 12 % of operations, and both infection and failure occur together in 0.78 % of operations. What percentage, P , of these operations succeed and are free from infection? Please round your answer to the nearest two decimal places. P = %

Answer 1

Answer:

The probability of not having an infection and the surgery being a success is 0.8478.

Step-by-step explanation:

Hello!

There are several possible outcomes facing surgery to repair a torn tendon:

"Infection" ⇒ P(In)= 0.04

"Repair fail" ⇒ P(F)= 0.12

"both infection and failure" ⇒ P(In∩F)= 0.0078

P(NIn∩S)=?

Where NIn represents "no infection" and S represents "successful surgery"

The event "no infection" is complementary to the event "infection" and so is it's probability P(NIn)= 1 - P(In)= 1 - 0.04= 0.96

The event "successful surgery" is complementary to the event "Repair fail" and so is it's probability: P(S)= 1 - P(F)= 1 - 0.12= 0.88

                         "Repair fail"      ;  "successful surgery";  Total

"Infection"       :   P(In∩F)           ;         P(In∩S)               ;     P(In)

"no infection" :  P(NIn∩F)         ;         P(NIn∩S)            ;     P(NIn)

                                P(F)            ;                P(S)              ;         1

P(F)   = P(In∩F) + P(NIn∩F)  

P(NIn∩F)  =   P(F)  - P(In∩F) = 0.12 - 0.0078= 0.1122

P(NIn∩S)= P(NIn) - P(NIn∩F) = 0.96-0.1122= 0.8478

The probability of not having an infection and the surgery being a success is 0.8478.

I hope it helps!