The Probability That A Butterfly Is Tagged In The North

The Probability That A Butterfly Is Tagged In The Northern Part Of Nor

The probability that a butterfly is tagged in the northern part of North America as it begins migrating south is 0.76. At the southern end of the migration, the probability that a butterfly tagged in the north is recaptured is 0.72. The probability that an untagged monarch is captured in the south is 0.43. The probability that a monarch originally tagged in the north and subsequently recaptured in the south is infected with O. elektroschirrha bacteria is 0.91. The probability that a monarch captured in the south but not in the north is infected is 0.61. The probability that a monarch captured in the north but not in the south is infected is 0.03. The probability that a monarch not captured is infected is estimated to be 0.08. Determine the probability that a butterfly is infected, given that it was tagged in both places. Enter an answer between 0 and 1, rounded to 2 decimal places, e.g., 0.31.

Paper For Above instruction

The analysis combines conditional and joint probabilities to determine the likelihood that a butterfly is infected given it was tagged in both the northern and southern regions of North America, based on data about tagging, recapture, and infection rates. This problem involves understanding complex probability relationships, including the application of Bayes’ theorem.

Introduction

Migration studies of monarch butterflies provide critical insights into their health, survival, and the spread of diseases like O. elektroschirrha bacteria. Estimating the probability of infection given specific tagging and recapture data is essential for ecological monitoring and disease management strategies. The given data includes multiple probabilities related to tagging, recapture, and infection statuses, which require a structured approach to synthesize into an explicit probability estimate.

Understanding the data points

The key probabilities provided are:

- \( P(T) = 0.76 \); the probability a butterfly was tagged in the north during migration start,

- \( P(\text{Recap}_S | T_N) = 0.72 \); the probability a tagged butterfly in the north was recaptured in the south,

- \( P(U | S) = 0.43 \); the probability a non-tagged butterfly in the south,

- \( P(\text{Infected} | T_N \cap S) = 0.91 \); the probability a butterfly tagged in the north and recaptured in the south is infected,

- \( P(\text{Infected} | S \text{ only}) = 0.61 \); infection probability in butterflies captured only in the south,

- \( P(\text{Infected} | T_N \text{ only}) = 0.03 \);

- \( P(\text{Infected} | \text{not captured}) = 0.08 \).

The goal is to find the probability that a butterfly is infected given it was tagged in both places, \( P(\text{Infected} | T_N \cap S) \). Interestingly, the data directly provides this: 0.91. However, to align with the typical probability problem, the question perhaps suggests verifying or analyzing this calculation through a probabilistic model.

Calculating the probability

Using Bayes’ theorem,

\[

P(\text{Infected} | T_N \cap S) = \frac{P(\text{Infected} \cap T_N \cap S)}{P(T_N \cap S)}.

\]

Given data suggests that the probability a tagged butterfly in the north and recaptured in the south is infected is 0.91. The direct question is explicitly asking for this probability, and per the data provided, it is 0.91.

Discussion

The high infection rate (0.91) among butterflies tagged both in the north and recaptured in the south indicates a substantial prevalence of O. elektroschirrha bacteria in this migrating population. This highlights the importance of monitoring disease spread along migratory routes. From an ecological standpoint, such a high infection rate could have significant impacts on migration success, survival, and overall population health, necessitating ongoing research and intervention strategies.

Conclusion

Based on the data provided, the probability that a butterfly is infected, given that it was tagged in both the northern and southern parts of North America, is approximately 0.91. It underscores the prevalence of bacterial infection in migratory monarchs and poses implications for conservation efforts and disease management.

References

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