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The problem involves determining the probability of getting **two or fewer women** when **19 people are hired**, assuming there is no discrimination and the number of men and women in the applicant pool is approximately equal.

### Key information:
- Total hires = 19
- Only 2 women were hired
- Probability of hiring a woman in a fair scenario (no gender discrimination) = 0.5
- We need to find \( P(\text{at most 2 women}) \) under the binomial distribution assumption.

### Step 1: Apply the binomial distribution formula
The binomial distribution is given by:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( n \) is the total number of trials (in this case, 19 hires)
- \( k \) is the number of successful outcomes (number of women hired)
- \( p \) is the probability of success in each trial (0.5, since the probability of hiring a woman is 50%)
- \( \binom{n}{k} \) is the binomial coefficient

We are interested in \( P(X \leq 2) \), i.e., the probability of hiring 2 or fewer women.

### Step 2: Calculate \( P(X \leq 2) \)
This means we need to sum the probabilities for \( X = 0 \), \( X = 1 \), and \( X = 2 \).

\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]

Using the binomial distribution:

\[
P(X = 0) = \binom{19}{0} (0.5)^0 (0.5)^{19} = \frac{1}{524288}
\]

\[
P(X = 1) = \binom{19}{1} (0.5)^1 (0.5)^{18} = \frac{19}{524288}
\]

\[
P(X = 2) = \binom{19}{2} (0.5)^2 (0.5)^{17} = \frac{171}{524288}
\]

So,

\[
P(X \leq 2) = \frac{1 + 19 + 171}{524288} = \frac{191}{524288} \approx 0.0003643
\]

### Step 3: Interpretation
The probability of hiring two or fewer women out of 19 employees is approximately **0.0003643**, or **0.03643%**.

### Step 4: Conclusion on Discrimination
Since the cutoff for unusual events is given as 0.5% (or 1 in 200), and the computed probability (0.03643%) is much lower than this, it suggests that hiring 2 or fewer women out of 19 is highly unlikely under fair hiring practices. Thus, **this supports a charge of gender discrimination**.

### Do you need further details or have any questions?

#### Related questions:
1. How does the binomial distribution apply to real-world scenarios like this?
2. What is the significance of the chosen 0.5% cutoff for unusual events?
3. Could a different probability cutoff alter the conclusion about discrimination?
4. How would the results change if more than two women were hired?
5. What are some potential biases not accounted for by this statistical model?

#### Tip:
When dealing with probabilities, always ensure the context justifies using specific distributions or cutoffs for decision-making.

After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only two women among the last 19 new employees. She also learns that the pool of applicants is very large, with an approximately equal number of qualified men as qualified women. Help her address the charge of gender discrimination by finding the probability of getting two or fewer women when 19 people are hired, assuming that there is no discrimination based on gender. (Report answer accurate to 8 decimal places).

This problem involves calculating the probability of hiring two or fewer women out of 19 employees, assuming no gender bias. Using binomial distribution, we determine whether such a low probability supports a claim of gender discrimination. The solution applies the binomial probability formula and compares the result against a 0.5% significance threshold.

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