Suppose that accidents occurring on a highway each day can be assumed to be weakly dependent (essentially independent) and that the probability of an accident occurring is small. Assume that the random variable \(X\) corresponding to the number of accidents occurring today has an expected value of three. What is the probability that exactly 2 accidents occur, given that it is known at least one accident occurs today?
\(X \sim Pois(\lambda = 3) \Rightarrow \mathbb{P}(X = x) = e^{-\lambda} \left(\dfrac{\lambda^x}{x!}\right) = e^{-3} \left(\dfrac{3^x}{x!}\right)\)
Want to find \(\mathbb{P}[ (X = 2) \,|\, (X \ge 1)]\)
Using the definition of conditional probability,
\[\mathbb{P}[ (X = 2) \,|\, (X \ge 1)] = \dfrac{\mathbb{P} [ (X = 2) \text{ AND } (X \ge 1) ]}{\mathbb{P}(X \ge 1)} = \dfrac{\mathbb{P}(X = 2)}{\mathbb{P}(X \ge 1)}\]
\(= \dfrac{\mathbb{P}(X = 2)}{1 - \mathbb{P}(X = 0)} = \dfrac{e^{-3}\left( \dfrac{3^2}{2!}\right)}{1 - \left[e^{-3}\left( \dfrac{3^0}{0!}\right)\right]} \approx \dfrac{0.224}{0.950} = 0.236\)
Question of interest: are females unfairly discriminated against in terms of promotions given by male supervisors?
Gender | promote | nopromote |
---|---|---|
Male | 21 | 3 |
Female | 14 | 10 |
\[ \mathbb{P}(\text{promote}\,|\,M) = 21/24 = 0.875 \\ \mathbb{P}(\text{promote}\,|\,F) = 14/24 = 0.583 \]
At a first glance, does there appear to be a relationship between promotion and gender?
We saw a difference of almost 30% (29.2% to be exact) between the proportion of male and female files that are promoted. Based on this information, which of the below is true?
\(H_0\), Null Hypothesis: "There is nothing going on."
\(H_a\), Alternative Hypothesis: "There is something going on.”