The probability that the shipment came from Ghana
"P(G) =0.3, P(G^c) =0.7"
Ineffective vials
"P(I|G) =0.08, P(I|G^c) =0.03"
n=30
The probability of 1 ineffective vial
"P(I=1|G)={30\\choose 1}0.08^1 0.92^{29}"
=0.2138
"P(I=1|G^c)={30\\choose 1}0.03^1 0.97^{29}"
=0.3721
The probability of getting an ineffective vial
"P(I=1) =P(I=1|G) P(G) +P(I=1|G^c) P(G^c)"
=0.2138*0.3+0.3721*0.7
=0.3246
The probability that the shipment came from neighbouring West African countries is calculated by applying the Bayes theorem
"P(G^c|I=1) =\\frac {P(I=1|G^c)P(G^c)}{P(I=1)}"
="\\frac {0.3721*0.7}{0.3246}"
=0.8024
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