We’re now going to look at the probability of hitting an area. This is a bull’s eye of sorts we could be playing darts or archery or just throwing a rock at a circle. It doesn’t really matter. But basically what the question is what’s the probability that whatever we throw is going to hit the red area.
The formula is still going to be the same as finding the probability of a numeric system. What we’re concerned with is the probability of the event we’re concerned with the number of events we’re concerned with over the event in the entire sample space. The number in this case is just going to be area.
What we’re really concerned with is the area that we want which is red, over the area of the entire thing. We have the different radius for each segment and so our entire area is just going to be the largest radius, Pi r² so we end up with an area of 9Pi. Our entire area is 9Pi.
We then want to find the area of just the red region. And so if this red region was solid what we would end up with is just the Pi r² for that, 2², 4Pi. But that is going to consider the entire red thing not just the doughnut that we’re left with so what we have to do is subtract off the center area, radius of 1 so that has an area of 1² Pi, just Pi. The larger red circle has an area 4Pi minus off the inner circle leaves us with an area of 3Pi for just this little red shell.
The area that we’re concerned with is 3Pi, simplifying this down we end up with 1/3. Probability of hitting red is just the area of red over the entire area, being able to simplify it down we end up with a probability of 1/3.