Probability calculation of events that are related in a non-standard form.

This should be a simple enough problem, but I find it to be a bit tricky. I am curious what the forums suggestions will be..

Look at the attached diagram.

I have the red curve, for which I have associated probabilities at each point, i.e. I know what P(x) and P(y) are numerically. Typical values for P(x)~0.1, P(y)~0.002.

This measurement (red curve) tells me that there is a star that lies somewhere on this red curve, but I don't know exactly where.

I have another curve (black) that intersects the measurement curve (red) at two points. The star can only be at one of the intersecting points, but the curve is standard, so it will intersect the curve at two points regardless.

The Q is: what is the probability of the black curve to intersect the red curve at x & y. This is the same question as "What is the probability of the star being at x or y".

A and A' are mutually exclusive/complementary events, i.e. the star is either at X or at Y.

A: the star is at X
A': the star is at Y

Can this be easily calculated?

Thanks in advance,


TS Contributor
Do OP means the following:

The location of star is randomly distributed on the red curve. Conditional on the star must be either located at x or y, what is the probability that the star is found at x and y respectively?
BGM is correct. A little more detail: I know from other constraints the probabilities of the 'star' on the red curve, e.g. P(x)=0.1, P(y)=0.002. There are in fact a set of black curves that intersect the red line at two points. I am trying to find the 'best' black curve that gives the highest probability.