The breakdown of workers in a particular state according to their political affiliation and type of job held is
shown here. Suppose a worker is selected at random within the state and the worker's political affiliation and
type of job are noted. Given the worker is a Democrat, what is the probability that the worker is in a white collar
job.
Political Affiliation
Republican Democrat Independent
White collar 15% 19% 7%
Type of job
Blue Collar 18% 10% 31%
The probability that the worker is in a white collar job is 19%.
To find the probability that a randomly selected worker, who is a Democrat, is in a white collar job, we need to use conditional probability.
Conditional probability is calculated using the formula:
P(A|B) = P(A ∩ B) / P(B)
Where:
P(A|B) is the probability of event A given that event B has already occurred.
P(A ∩ B) is the probability of both events A and B occurring together.
P(B) is the probability of event B occurring.
In this case, event A is being a Democrat and event B is having a white collar job.
According to the given information, the probability of a worker being a Democrat and having a white collar job (P(A ∩ B)) is 19%.
The probability of a worker being a Democrat (P(B)) is the sum of the probabilities of being a Democrat and having a white collar job, and being a Democrat and having a blue collar job. So in this case, P(B) = 19% + 10% = 29%.
Now we can calculate the conditional probability:
P(A|B) = P(A ∩ B) / P(B)
P(A|B) = 19% / 29%
P(A|B) ≈ 0.655 or 65.5%
Therefore, given that the worker is a Democrat, the probability that the worker is in a white collar job is approximately 65.5%.
To find the probability that the worker is in a white collar job given that they are a Democrat, we need to use conditional probability.
The probability of a worker being a Democrat and in a white collar job is 19%.
The probability of a worker being a Democrat is (19% + 10% + 31%) = 60%.
Using the formula for conditional probability:
P(White collar job | Democrat) = P(White collar job and Democrat) / P(Democrat)
P(White collar job | Democrat) = 19% / 60% = 0.3167 or 31.67%.
Therefore, the probability that a worker is in a white collar job given that they are a Democrat is 31.67%.