6.3 (10 points) In a company there are 7000 employees and 2000 of them have been

Question

6.3 (10 points)

In a company there are 7000 employees and 2000 of them have been with the company for less than 3 years. The salary of an employee is uniformly distributed between $50, 000 and $70, 000 if the employee has been with the company for less than 3 years. The salary is uniformly distributed between $60, 000 and $100, 000 if the employee has been with the company for 3 years or longer.

• If you know that Mr. Smith’s salary is between $60, 000 and $70, 000 what is the probability that Mr. Smith has been with the company for less than 3 years?

• If you know that Mr. White’s salary is between $65, 000 and $75, 000 what is the probability that Mr. White has been with the company for less than 3 years?

• If you know that Mr. Green’s salary is between $65, 000 and $75, 000 what is the probability that Mr.Green has been with the company for less than 3 years?

Explanation / Answer

P($60,000 to $70,000 salary for someone who has been with the company for less than 3 years)

= (70,000-60,000)/(70,000-50,000)

= 1/2

P($60,000 to $70,000 salary for someone who has been with the company for more than 3 years)

= ((70,000-60,000)/(100,000 - 60,000)

= 1/4

P(less than 3 years | salary between $60,000 and $70,000) = P(less than 3 years and salary between $60,000 and $70,000)/P(salary is between  $60,000 and $70,000)

= (2000/7000)x1/2/((2/7 x 1/2) + (5/7 x 1/4))

= 0.4444

If you know that Mr. Smith's salary is between $60,000 and $70,000 the probability that Mr. Smith has been with the company for less than 3 years = 0.4444

_______________________________________________________________________________________________

P($65,000 to $75,000 salary for someone who has been with the company for less than 3 years)

= (70,000-65,000)/(70,000-50,000)

= 1/4

P($65,000 to $75,000 salary for someone who has been with the company for more than 3 years)

= ((75,000-65,000)/(100,000 - 60,000)

= 1/4

P(less than 3 years | salary between $65,000 and $75,000) = P(less than 3 years and salary between $65,000 and $75,000)/P(salary is between  $65,000 and $75,000)

= (2/7)x1/4 / ((2/7 x 1/4) + (5/7 x 1/4))

= 2/7

=0.2857

If you know that Mr. White's salary is between $65,000 and $75,000 the probability that Mr. White has been with the company for less than 3 years = 0.2857

If you know that Mr. Green's salary is between $65,000 and $75,000 the probability that Mr. Green has been with the company for less than 3 years = 0.2857

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