Mr James Bond takes his car to the mechanic for regular servicing. The mechanic runs a test on the car transmission. The test would return one of two values: $1($Test finds Fault$)$ or $0($test finds No Fault$).$ The accuracy of the test is as follows: The probability of the test returning $1$ when the car transmission is actually faulty is $0\mathrm{.}99,$ and the probability of the test returning $0$ when the car transmission is NOT faulty is $0\mathrm{.}97.$ After running the test on Mr James Bond$$s car, the test returns $1.$ According to the manufacturer of Mr James Bond$$s car, at the age of his car, only $1$ in $500$ cars would have a faulty transmission.

Please use the following vocabulary when answering the following questions using Bayes$$ rules: $($Your answer will need to include the working out of how you derive the answer.$)$

F $$ The car transmission is actually faulty;

T$1$ The test returns the value $1.$