A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials.

For Company X’s shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective.

The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective.

Calculate the probability that this shipment came from Company X.

• 0.10
• 0.14
• 0.37
• 0.63
• 0.86

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

Let

[math]C[/math] = Event that shipment came from Company X

[math]I_1[/math] = Event that one of the vaccine vials tested is ineffective

Then by Bayes’ Formula,

[[math]] \operatorname{P}[C | I ] = \frac{\operatorname{P}[ I_1 | C ] \operatorname{P}[ C ]}{\operatorname{P}[ I_1 | C ] \operatorname{P}[ C ] + \operatorname{P}[ I_1 | C^c ] \operatorname{P}[C^c ]}. [[/math]]

Now

[[math]] \begin{align*} \operatorname{P}[C] &= 1/5 \\ \operatorname{P}[C^c] &= 1-1/5 = 4/5 \\ \operatorname{P}[ I_1 | C ] &= \binom{30}{1}(0.1)(0.9)^{29} = 0.141 \\ \operatorname{P}[I_1 | C^c] &= \binom{30}{1}(0.02)(0.98)^{29} = 0.334 \end{align*} [[/math]]

Therefore,

[[math]] \operatorname{P}[C | I_1] = \frac{(0.141)(1/5)}{(0.141)(1/ 5) + ( 0.334 )( 4 / 5)} = 0.096 [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.