During an international flight, a pilot notices a warning light come on somewhere on the panel. The light indicates that one of the landing wheels is...

During an international flight, a pilot notices a warning light come on somewhere on the panel. The light indicates that one of the landing wheels is not fully functional. The pilot reasons that this may mean that there is actually a problem with the wheel, however it may also mean that the warning light itself is malfunctioning. There are back-up wheels for this sort of situation, but the pilot would still like to know whether or not the wheel is fully functional.

The pilot, who is very experienced with the mechanical and technical specifications of the plane, knows the following probabilities:

• There is only ever a problem with a wheel in 1 out of every 1,000 flights. That is, the probability of there being a problem with a wheel on any given flight is 0.001.
• The warning light is very accurate when it comes to positively reporting a problem. Specifically, if thereisa problem with one of the wheels, the probability that the warning light will come on is 1.
• However, the warning light also occasionally comes on even if there is no problem - although this is extremely rare. The probability that the light will come on during a flight when thereisn'ta problem is 0.0002.

Based on this information, calculate the probability that there is a problem with a wheel given that the warning light has just come on for the pilot. Give your answer as a decimal to 3 decimal places.

P(there is a problem|warning light is on) =