6. The Flu season is coming up soon. Suppose a small tech company with 300 emplo

Question

6. The Flu season is coming up soon. Suppose a small tech company with 300 employees obtains its flu vaccines from 2 different medical companies (say GS and PC). Assume that from past studies they have found that GS is 70% effective whilst PC is 55 % effective (independent across employees). Also assume that GS costs the company $30 per vaccine whilst PC costs the tech company S15 per vaccine. Finally assume that if a person falls ill from the flu then this costs the company $300 per person in productivity loss. (a) 1 point| Suppose all 300 employees get vaccinated. Assume that the company got all the vaccines from GS. Let C denote the total cost to the company owing to the flue season (here C is the sum of the total cost of buying the vaccines + cost incurred owing to employees getting sick despite taking the vaccine). Compute E(C) and Var(C) (b)2 points Brad belongs to a smaller group called the "Applied probability models" group in the company which has 8 people (including himself). Suppose Brad does not know if the company got the vaccine from company GS or PC. So his prior probability is P(GS) .5 and P(PC) .5. Now he notices that in his group of 8, 4 people have fallen ill with the flu (note: by assumption all 8 have taken the flu shot). Let S4 be the above event namely 4 out of 8 people have fallen ill with the flu. Compute P(PC|S4).

Explanation / Answer

(a) Number of eployees = 300

All vaccines are from GS company which 70 % effective that means 30% ineffective and costs $ 30 per patient

here C = Cost of buying vaccine + Cost of employee being sick despite taking the vaccine

C = 300 * 30 + 300 * X = 9000 + 300 X

where X is the number of employees got sick due to productivity loss.

so E(C) = E(9000 + 300 X) = 9000 + 300 E(X)

where X ~ BIn (X; 300; 0.3)

E(X) = 300 * 0.3 = 90

E (C) = 9000 + 90 * 300 = $ 36000

Var(C) = Var (9000 + 300X) = 3002 Var (X)

Var(X) = (0.3 * 0.7 * 300) = 63

Var(C) = 5670000

(b) 4 people have fallen ill with the flu

S4 = 4 out of the 8 people have fallen ill with the flu.

Here calculate P(PC l S4 ) = P ( PC and S4)/ P (S4) = P (S4 l PC) P(PC) / P(S4 )

P (any random employee will fall ill) = P (GS) * P(S4 l GS) + P(PC) * P(S4 l PC) = 0.5 * 0.7 + 0.5 * 0.55= 0.625

P ( 4 out of 8 employee will fall ill) = 8C4 P (GS) * P(S4 l GS) +  8C4 P(PC) * P(S4 l PC)

= 8C4 * 0.5 * [ (0.7)4 (0.3)4 + (0.55)4 (0.45)4]

= 0.1993

P(S4 ) = 0.1993

now, P(PC) = 0.5

P(S4 and  PC) = 8C4 * 0.5 (0.7)4 (0.3)4 = 0.068

so P(PC l S4 ) = 0.068/ 0.1993 = 0.3415

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