A hospital obtains 40% of its flu vaccine from Company A, 50% from Company B, and 10% from Company C. From past experience, it is known that 3% of the vials from A are ineffective, 2%from B are ineffective, and 5% from C are ineffective. The hospital tests five vials from each shipment. If at least one of the five is ineffective, find the conditional probability of that shipment’s having come from C.

Solution 15E

Step1 of 4:

We have,

Hospital obtains vaccine from company a = A

Hospital obtains vaccine from company b = B

Hospital obtains vaccine from company c = C

One of the 5 vials is ineffective = E0

We need to find the conditional probability of that shipment’s having come from C.

Step2 of 4:

From the given information we have,

P(A) = 40%

= 0.4

P(B) = 50%

= 0.5

P(C) = 10%

= 0.1

Similarly,

We have

P(E/A) = [3%]

= [0.03]

= 0.030.8852

= 0.0265

Hence, P(E/A) = 0.0196

P(E/B) = [2%]

= [0.02]

= 0.020.9223

= 0.0184

Hence, P(E/B) = 0.0184

P(E/C) = [5%]

= [0.05]

= 0.050.8145

= 0.0407

Hence, P(E/C) = 0.0407

Step3 of 4:

Now,

We need to find the probability of ineffective vials and it is given by

P(E) = P(EA) + P(EB) + P(EC) ………(1)

Where,

1).P(EA) = P(A)P(E/A) [Because P(E/A) = ]

= 0.40.0265

...