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Answer :
The overall probability of an outbreak from chicken pathogens transmitted to humans is approximately 0.3916.
To find the overall probability of an outbreak, we can use the law of total probability, which states:
[tex]\[ P(\text{Outbreak}) = P(\text{Outbreak} | \text{Type 1}) \times P(\text{Type 1}) + P(\text{Outbreak} | \text{Type 2}) \times P(\text{Type 2}) \][/tex]
Given:
- [tex]\( P(\text{Outbreak} | \text{Type 1}) = 0.51 \)[/tex]
- [tex]\( P(\text{Outbreak} | \text{Type 2}) = 0.19 \)[/tex]
- [tex]\( P(\text{Type 1}) = 0.63 \)[/tex]
- [tex]\( P(\text{Type 2}) = 0.37 \)[/tex]
Let's calculate:
1. Probability of outbreak given Type 1:
[tex]\[ P(\text{Outbreak} | \text{Type 1}) = 0.51 \][/tex]
2. Probability of outbreak given Type 2:
[tex]\[ P(\text{Outbreak} | \text{Type 2}) = 0.19 \][/tex]
3. Overall probability of outbreak:
[tex]\[ P(\text{Outbreak}) = (0.51 \times 0.63) + (0.19 \times 0.37) \][/tex]
Now, let's compute this:
[tex]\[ P(\text{Outbreak}) = (0.51 \times 0.63) + (0.19 \times 0.37) \][/tex]
[tex]\[ P(\text{Outbreak}) = (0.3213) + (0.0703) \][/tex]
[tex]\[ P(\text{Outbreak}) = 0.3916 \][/tex]
So, the overall probability of an outbreak is approximately [tex]\( 0.3916 \)[/tex] when rounded to five decimal places.
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