A retailer currently employs 22 sales associates, of which 8 are more than 55 years in age. Due to the recent downturn in the economy, the retailer decided to reduce its sales force by 7 employees. Of the 7 people laid off, 5 were older employees ( 55 years in age). If the employees who were laid off were randomly selected, what is the probability that 5 of them would be from the older group?

Question

A retailer currently employs 22 sales associates, of which 8 are more than 55 years in age. Due to the recent downturn in the economy, the retailer decided to reduce its sales force by 7 employees. Of the 7 people laid off, 5 were older employees (> 55 years in age). If the employees who were laid off were randomly selected, what is the probability that 5 of them would be from the older group?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We need to find the probability that 5 out of 7 randomly selected laid-off employees are from the older group. This can be modeled using the hypergeometric distribution. ‖ ‖ step 2 ⋮ The hypergeometric distribution formula is given by: $$ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} $$ where: - $ N $ is the total number of items (22 employees), - $ K $ is the number of success states in the population (8 older employees), - $ n $ is the number of draws (7 employees laid off), - $ k $ is the number of observed successes (5 older employees laid off). ‖ ‖ step 3 ⋮ Calculate the binomial coefficients: $$ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = 56 $$ $$ \binom{14}{2} = \frac{14!}{2!(14-2)!} = \frac{14!}{2!12!} = 91 $$ $$ \binom{22}{7} = \frac{22!}{7!(22-7)!} = \frac{22!}{7!15!} = 170544 $$ ‖ ‖ step 4 ⋮ Substitute these values into the hypergeometric formula: $$ P(X = 5) = \frac{\binom{8}{5} \binom{14}{2}}{\binom{22}{7}} = \frac{56 \times 91}{170544} = \frac{5096}{170544} \approx 0.0299 $$ ‖ ∻Answer∻ ⚹ The probability that 5 out of 7 randomly selected laid-off employees are from the older group is approximately 0.0299. ⚹ ∻Key Concept∻ ⚹Hypergeometric Distribution⚹ ∻Explanation∻ ⚹The hypergeometric distribution is used to calculate the probability of a specific number of successes in a sample drawn without replacement from a finite population.⚹◊What is the formula for calculating the probability of selecting 5 out of 7 employees from a group of 8 older employees randomly for layoff⍭ Generate me a similar question◊

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