# Probability of C occurring given both A and B have occurred?

The problem is as follows:

Tags are used to classify images. An image with the tag "sky" has 90% chance of containing a sunset, with the tag "cloud" giving an image 50% chance of containing a sunset. Given an image with both tags, what is the likelihood that this image contains a sunset? The addition of extra tags should increase the likelihood, i.e. the probability will be at least equal to the highest tag (in this case 90%).

In general:

Given the probability of C occurring given A is x, and the probability of C occurring given B is y, what is the probability of C occurring given both A and B have occurred?

The general solution to this seems like something that should be somewhat obvious and easily searchable, but my mind is drawing a blank on what to search for here.

• Logical observation: If all cloud images are sky images, and half of all cloud images are sunset images, then half of all cloud-sky images are sunset images. It seems clear that more information is needed. – Logophobic Jan 28 '16 at 11:54
• I suppose having tag "cloud" and tag "sky" are not independent events, as clouds usually appear in the sky (except e.g. computing clouds). Also I doubt more tags should always increase the probability in this case: moonlit night images may well depict clouds in the sky without a sunset, so additional tag "night" shoud decrease the probability in question... – CiaPan Aug 22 '17 at 10:34

If there is independence then the probability of no sunset is: $$\frac{10}{100}\frac{50}{100}=\frac1{20}$$
A priori there are $8$ different cases: $s\in\{0,1\}$, $c\in\{0,1\}$, and $\odot\in\{0,1\}$ with unknown dependencies between $s$, $c$, and $\odot$. You have to find out whether the available data suffice to determine the eight probabilities $p_{s\,c\,\odot}$. If yes it is then easy to compute the conditional probability you are looking for.