To determine the probability that a microbial contamination in capsule A is from salmonella, we need to follow the thought process of breaking down the problem through conditional probability.
The probabilities given for Capsule A are as follows:
- Microbial contamination from E. Coli: [tex]\(0.15 \%\)[/tex]
- Microbial contamination from Staphylococcus: [tex]\(0.28 \%\)[/tex]
- Microbial contamination from Pseudomonas: [tex]\(0.33 \%\)[/tex]
- Microbial contamination from Salmonella: [tex]\(0.31 \%\)[/tex]
First, we need to calculate the total probability of microbial contamination in Capsule A. This is the sum of all individual probabilities of microbial contaminations:
[tex]\[
\text{Total microbial contamination} = 0.15\% + 0.28\% + 0.33\% + 0.31\%
\][/tex]
Let's add these probabilities:
[tex]\[
\text{Total microbial contamination} = 1.07\%
\][/tex]
Now that we have the total probability of microbial contamination, we can determine the chance that the contamination is specifically from salmonella. This is found by dividing the probability of contamination from salmonella by the total probability of microbial contamination:
[tex]\[
\text{Chance of salmonella} = \frac{\text{Probability of Salmonella contamination}}{\text{Total microbial contamination}}
\][/tex]
Substitute in the known values:
[tex]\[
\text{Chance of salmonella} = \frac{0.31\%}{1.07\%}
\][/tex]
[tex]\[
\text{Chance of salmonella} \approx 0.2897
\][/tex]
Converting this probability to a percentage:
[tex]\[
0.2897 \approx 28.97\%
\][/tex]
Hence, if capsule A shows microbial contamination, the chance that the contamination is from salmonella is approximately [tex]\(28.97\%\)[/tex].
