The joint distributions of voter ages and party preferences in a certain city is described
by the following table:
Age/Party conservative liberal undecided Total
under 30 0.05 0.095 0.155 0.30
30-50
over 50
total 1
How to calculate the probability that the randomly selected voter is conservative.
The probability for each age,party pair is specified in the table.
I have just provided some values for reference
My Approach :
A randomly selected voter is conservative. Do I just add the probabilities in the conservative
column ?
There is one more variant which I want to address – probability that a randomly selected voter to
be conservative if under 30
Best Answer
For your first question, you're correct that you can simply add up the probabilities. A voter cannot be in more than one of your age brackets, so we can apply the law of total probability:
$$P(\text{conservative}) = \Sigma_i P(\text{conservative} \cap \text{age}_i)$$
Your second question refers to what's called a conditional probability, the probability of one event given another event. The formula for that is as follows:
$$P(\text{conservative}|\text{age} < 30) = \frac{P(\text{conservative} \cap \text{age} < 30)}{P(\text{age}<30)} = \frac{P(\text{conservative} \cap \text{age} < 30)}{\Sigma_i P(\text{age} < 30 \cap \text{party}_i)}$$
So essentially, you would add up the total probabilities for the row (as you have in the last column) and divide the probability in the cell for conservatives under thirty by that total probability.
Hope this helps!