This is a basic probability problem, and I can get the solution. But for a while I was using a wrong approach. My question for the forum is why my initial approach was wrong.
The problem: You're on a ship sailing off a coast in a 2-dimensional world, and to fix your position you take three lines of bearing. Each of the three readings is independent and has independent errors. Errors are symmetrical for each bearing, i.e. you're equally likely to get a bearing which is too high as a bearing which is too low. The three observed lines of bearing will form a triangle. What is the probability of your ship being inside the triangle.
The solution: The probability is 25%. To see this, consider each line of bearing as having equal probability of being too high or too low. You therefore have eight error possibilities: +++, ++-, +-+, etc. If you draw out all eight, you'll see that in two of them the ship is in the triangle formed. Hence, 25%.
My original, incorrect approach: I drew out three observed lines of bearing, and considered the possibility that the ship was in the triangle formed. However, the three lines divide the plane into only seven areas. In this approach, I considered the probability that the ship was "above" or "below" a given line of bearing as 50%. (As opposed to the correct approach, where the probability that an observed line was "above" or "below" the correct position of the ship was 50%.) But, like I said, the ship has only seven (not eight) possible combinations of "above" and "below" for the three lines of bearing -- one of those eight combinations is a geometrical impossibility. So this line of reasoning was not fruitful, and I couldn't get anywhere until I thought the correct way.
My question for the forum: Why was my original approach wrong? I'm thinking it has something to do with considering the position of the ship as the random variable rather than the errors of each bearing line as random variables. Or else, in drawing three bearing lines, I'm already excluding some probabilities (i.e. making a choice)?
I know I'm being fast and loose with my terminology, but I'm hoping that if you think of three bearing lines on a plane, my meanings are clear.