A long-suffering manager has two employees. On any day, Employee A has a 80% cha

Question

A long-suffering manager has two employees. On any day, Employee A has a 80% chance of showing up for work, while Employee B has a 90% chance of showing up for work. The probability that both Employee A and Employee B will show up is 75%.
a. Are the events “Employee A will show for work” and “Employee B will show for work” independent? Justify your answer using mathematical reasoning.
b. Knowing that only one employee showed for work in a given day, what is the probability that that Employee A showed for work on that day?
c. Give the probability distribution for X = number of employees that show up for work in a day. A long-suffering manager has two employees. On any day, Employee A has a 80% chance of showing up for work, while Employee B has a 90% chance of showing up for work. The probability that both Employee A and Employee B will show up is 75%.
a. Are the events “Employee A will show for work” and “Employee B will show for work” independent? Justify your answer using mathematical reasoning.
b. Knowing that only one employee showed for work in a given day, what is the probability that that Employee A showed for work on that day?
c. Give the probability distribution for X = number of employees that show up for work in a day. A long-suffering manager has two employees. On any day, Employee A has a 80% chance of showing up for work, while Employee B has a 90% chance of showing up for work. The probability that both Employee A and Employee B will show up is 75%.
a. Are the events “Employee A will show for work” and “Employee B will show for work” independent? Justify your answer using mathematical reasoning.
b. Knowing that only one employee showed for work in a given day, what is the probability that that Employee A showed for work on that day?
c. Give the probability distribution for X = number of employees that show up for work in a day.

Explanation / Answer

a) If these events are independent then:

P(A shows up and B shows up) = P(A shows up)*P(B shows up)

Now,

P(A shows up)*P(B shows up) = 0.80*0.90 = 0.72 which is not equal to P(A shows up and B shows up). Hence,

These events are not independent.

b) P(Only A shows up) = 0.80 = 0.75 = 0.05

P(Only B shows up) = 0.90 - 0.75 = 0.15

Hence,

P(Only one shows up) = 0.15 + 0.05 = 0.20

Therefore,

P(A showed up | Only one showed up) = 0.05/0.20 = 5/20 = 1/4 = 0.25

c) The probability distribution will be:

x p(x) 0 1 - (0.80 + 0.90 - 0.75) = 1 - 0.95 = 0.05 1 0.15 + 0.05 = 0.20 2 0.75

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