|Now that we know the basic probabilities of individual tags, we can also find the joint probabilities of some of these events. For instance, how often do a deciduous tree and a coniferous tree appear in the same painting? We know that 57 percent of paintings contain a deciduous tree and 53 percent of paintings contain a coniferous tree. According to our data set, 20 percent of paintings contain at least one of each.
What’s more, we can also find the probability that Ross painted something given that he painted something else, a statistic that’s called conditional probability.
Conditional probability can be a bit tricky. We know that 44 percent of Ross’s paintings contain clouds, 9 percent contain the beach and 7 percent contain both the clouds and the beach. We can use this information to figure out two things: the probability that Ross painted a cloud given that he painted a beach, and the probability that he painted a beach given that he painted a cloud. You divide the joint probability — 7 percent in this case — by the probability of the given — 44 percent or 9 percent, depending on whether you want to know the probability of a beach given a cloud or a cloud given a beach.
The biggest pitfall people often face is assuming the two probabilities are the same. The probability that Ross painted a cloud given that he painted the beach — essentially, how many beach paintings have clouds — is (0.07)/(0.09), which is 78 percent. The vast majority of beach scenes contain clouds. However, the probability that Ross painted a beach given that he painted a cloud — or, how many cloud paintings contain a beach — is (0.07)/(0.44), or 16 percent. So the vast majority of cloud paintings don’t have beaches.
I figured out the conditional probability of every Bob Ross tag against every other tag to answer the following pressing questions.
What is the probability, given that Ross painted a happy tree, that he then painted a friend for that tree?
There’s a 93 percent chance that Ross paints a second tree given that he has painted a first.
What percentage of Bob Ross paintings contain an almighty mountain?
About 39 percent prominently feature a mountain.
What percentage of those paintings contain several almighty mountains?
Ross was also amenable to painting friends for mountains. Sixty percent of paintings with one mountain in them have at least two mountains.