Answer:
The probability that a promoted officer is female is 0.1111, or approximately 11.11%.
Step-by-step explanation:
We can use the given data to construct a contingency table as follows:
Male Female Total
Promoted 288 36 324
Not Promoted 672 204 876
Total 960 240 1200
a) The probability of getting promoted within the department is the proportion of officers who were promoted, which is 324/1200 = 0.27, or 27%.
b) Given that a randomly selected officer was promoted, we need to find the probability that the selected officer is female. This is a conditional probability and can be calculated using Bayes' theorem:
P(Female | Promoted) = P(Promoted | Female) * P(Female) / P(Promoted)
where P(Promoted | Female) is the probability of being promoted given that the officer is female, P(Female) is the probability of selecting a female officer, and P(Promoted) is the overall probability of being promoted.
We can calculate each of these probabilities as follows:
P(Promoted | Female) = 36/240 = 0.15
P(Female) = 240/1200 = 0.2
P(Promoted) = 324/1200 = 0.27
Substituting these values into the equation gives:
P(Female | Promoted) = 0.15 * 0.2 / 0.27 = 0.1111
So the probability that a promoted officer is female is 0.1111, or approximately 11.11%.