Home $>2023$ Fall2 MAT120 (TTh11:30) > Assessment 3.1 - Binomial Distributions Score: $24.9 / 40 \quad 12 / 16$ answered Question 10 After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only two women among the last 21 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 21 people are hired, assuming that there is no discrimination based on gender. You may answer as a decimal or in scientific notation, but provide at least three significant digits in your answer. $P($ at most two $)=$ Because this is a serious claim, we will use a stricter cutoff value for unusual events. We will use $0.5 \%$ as the cutoff value ( 1 in 200 chance of happening by chance). With this in mind, does the resulting probability really support such a charge? no, this does not support a charge of gender discrimination yes, this supports a charge of gender discrimination Question Help: $D$ Post to forum Submit Question

Step 1 ：Given that the Bellevue Credit Company has hired only two women among the last 21 new employees, we are to find the probability of getting two or fewer women when 21 people are hired, assuming that there is no discrimination based on gender.

Step 2 ：We use the binomial probability formula to calculate this probability. The binomial probability formula is given by \( P(k; n, p) = C(n, k) \cdot p^k \cdot (1-p)^{n-k} \), where \( C(n, k) \) is the binomial coefficient, \( n \) is the number of trials, \( k \) is the number of successful trials, and \( p \) is the probability of success on a single trial.

Step 3 ：In this case, \( n = 21 \), \( p = 0.5 \) (since we are assuming no gender discrimination, the probability of hiring a woman is the same as the probability of hiring a man), and we are interested in the cases where \( k = 0, 1, 2 \).

Step 4 ：We calculate the probabilities for \( k = 0, 1, 2 \) and sum them up to get the total probability of hiring two or fewer women.

Step 5 ：The calculated probabilities are approximately \( 4.77 \times 10^{-7} \), \( 1.00 \times 10^{-5} \), and \( 1.00 \times 10^{-4} \) for \( k = 0, 1, 2 \) respectively.

Step 6 ：The total probability is approximately \( 0.00011 \).

Step 7 ：We then compare this total probability with a cutoff value of \( 0.005 \) (which corresponds to a 1 in 200 chance of happening by chance) to determine if the result supports a charge of gender discrimination.

Step 8 ：Since the total probability is less than the cutoff value, this result supports a charge of gender discrimination.

Step 9 ：Final Answer: The probability of hiring two or fewer women when 21 people are hired, assuming that there is no discrimination based on gender, is approximately \(\boxed{0.00011}\). This result supports a charge of gender discrimination.