Probability: combining multiple probabilities to reach a composite - statistics

Question

Below I've included data from a PEW research study. What is the method for combining probabilities to reach a composite for say: an 18 year old black male?

Answer 1

As pointed out by Imran, one cannot deduce the answer from the limited data that are available. If you're willing to make a simplifying assumption, you can make progress. Note, however, that whether or not this assumption is valid can only be answered by getting more detailed data.
Here we go. OP is asking for P(cell phone|age, race, gender). By Bayes' rule, this is:
P(cell phone|age, race, gender)
= P(age, race, gender, cell phone) / P(age, race, gender)
= P(age, race, gender|cell phone) P(cell phone) / P(age, race, gender)
The simplifying assumption is that age, race, and gender are independent given cell phone status. Again, whether this is valid can't be answered with the available data. Assuming that, we have:
P(age, race, gender|cell phone)
= P(age|cell phone) P(race|cell phone) P(gender|cell phone)
Now apply Bayes' rule to each term:
P(age|cell phone) = P(cell phone|age) P(age) / P(cell phone)
P(race|cell phone) = P(cell phone|race) P(race) / P(cell phone)
P(gender|cell phone) = P(cell phone|gender) P(gender) / P(cell phone)
At this point we have:
P(age, race, gender, cell phone)
= P(cell phone|age) P(cell phone|race) P(cell phone|gender)
P(age) P(race) P(gender) / P(cell phone)^2
Let P1 = P(age, race, gender, cell phone) and P0 = P(age, race, gender, no cell phone). Then P(age, race, gender) = P1 + P0, and
P(cell phone|age, race, gender) = P1/(P1 + P0) = 1/(1 + P0/P1)
Now, happily, some terms cancel:
P0/P1 = foo/bar
with
foo = P(no cell phone|age) P(no cell phone|race) P(no cell phone|gender) / P(no cell phone)^2
bar = P(cell phone|age) P(cell phone|race) P(cell phone|gender) / P(cell phone)^2
Some examples:
P(cell phone|age = 18-29, race=black, gender=male)
= 1 / (1 + ((0 * 0.02 * 0.05) / 0.05^2) / ((1 * 0.98 * 0.95) / 0.95^2))
= 1
P(cell phone|age = 30-49, race=black, gender=male)
= 1 / (1 + ((0.02 * 0.02 * 0.05) / 0.05^2) / ((0.98 * 0.98 * 0.95) / 0.95^2))
= 0.992
P(cell phone|age = 65+, race=white, gender=female)
= 1 / (1 + ((0.15 * 0.06 * 0.06) / 0.05^2) / ((0.85 * 0.94 * 0.94) / 0.95^2))
= 0.794
So, there are some results. Again, remember that these results depend on an assumption that can only be verified with more data.

Answer 2

There is not enough information to determine exactly how many people in a combined group have a cell phone, because we don't know exactly how those groups overlap.
Let's consider a simpler example: Out of 100 people, 50 are men and 50 like cheese. How many are men who like cheese?
Clearly we don't have enough information, because anywhere from none of the men to all of the men could like cheese.
The same concept applies to the cell phone data, and furthermore it is difficult to even come up with ranges of possibilities.
For example consider how many hispanic men have cellphones. It should be between 95% and 98%, right? Wrong! Imagine there are 10k men in the survey , 990 hispanic women, but only 10 hispanic men. We could have 9.5k non-hispanic men, 980 hispanic women, and 0 hispanic men who have a cell phone - giving us 0% of hispanic men owning a cell phone. Or by similar reasoning we could construct a case where 100% of hispanic men own a cell phone.
However, if we have data on exactly how many of each group were surveyed you might be able come up with some possible ranges that are narrower than 0-100%. For example in the men who like cheese example if 60 of the people were men then we could say at least 10 must like cheese.