Probability of an attribute in one member of a population

#1
Wondering whether the following can be calculated (or whether more info is needed).

Here's the population:
A - 224,000,000 (72.5%)
B - 39,000,000 (12.6%)
C - 46,000,000 (14.9%)
TOTAL population - 309,000,000

In that population, members have a certain attribute X, as follows:
A - 232,000 (59.5%)
B - 147,000 (37.7%)
C - 11,000 ( 2.8%)
TOTAL with attribute X - 390,000

Questions
1. Since B having attribute X is disproportionately overrepresented compared to their representation in the total population, is it correct that one should expect a randomly selected member B to be more likely to have that attribute than member A?
2. What is the probability that a member of the population is both A and has the attribute X?
3. What is the probability that a member of the population is both B and has the attribute X?
4. If one randomly selected member of the population is A, what is the probability that member has attribute X?
5. If one randomly selected member of the population is B, what is the probability that member has attribute X?
6. If two members of the population are randomly selected, one A and one B, what is the probability that member A has attribute X?
6. If two members of the population are randomly selected, one A and one B, what is the probability that member B has attribute X?
 

hlsmith

Omega Contributor
#2
Well you have to put forward some type of effort in answering these questions. We won't just answer them for you.


So yes, these questions seem to have tangible answers. What direct questions do you have?
 
#3
Thanks for the response.

I'm brand new to this forum, so perhaps I've misunderstood its purpose. I'm not a student seeking answers for an assignment - I'm an adult who recently had a conversation about this topic among friends, none of whom could recall enough probability knowledge to answer the questions.

I'd be happy to calculate the probabilities myself if provided the formulas, or the logic needed to determine the result.

Or perhaps you can explain what additional effort is required of me, or describe what direct questions might be asked that are different from those listed in the original post?

Thanks!
 
#4
1. Prob that a randomly selected individual from sub-pop A has attribute X = P(X|A) = 232,000/224,000,000 = 0.00103571. P(X|B) = 147,000/39,000,000 = 0.00376923. Since P(X|B) > P(X|A), the answer is yes, a randomly selected member from B is more likely to have attribute X than one from A.
2. Prob that a randomly selected individual from the whole pop is from sub-pop A = P(A) = 224,000,000/(224,000,000+39,000,000+46,000,000) = 0.72491909. P(X & A) = P(X|A)*P(A) = 0.00103571*0.72491909 = 0.00075081.
3. Follows same logic as 2.
4. See 1.
5. See 1.
6. See 1. Why would selecting an additional member from B affect the A member's prob?
7. See 1. Why would selecting an additional member from A affect the B member's prob?

The problem description suggests many much more interesting questions with which you might like to play around. For example, what is the prob that of n randomly selected individuals from the whole pop, at least k are X and m are from B where nkm?
 
#5
Thanks a bunch! That really makes sense now that I look at what you did. About questions 6 and 7, you're correct that those scenarios don't affect the probabilities you provided earlier, though when I originally posted I thought they might.

Thanks again!