Suppose that a pest control program is dealing with two types of rats, black and brown.It is found that the probability of a rat being brown, given that it is caught in a trap, is 0.27.It is estimated that the probability of any rat being caught is 0.37 and thatthe probability of a randomly chosen rat in the region being brown is 0.58.What is the probability of a rat being caught in a trap given that it is brown?
Question
Suppose that a pest control program is dealing with two types of rats, black and brown.It is found that the probability of a rat being brown, given that it is caught in a trap, is 0.27.It is estimated that the probability of any rat being caught is 0.37 and thatthe probability of a randomly chosen rat in the region being brown is 0.58.What is the probability of a rat being caught in a trap given that it is brown?
Solution
To find the probability of a rat being caught in a trap given that it is brown, we can use Bayes' theorem. Bayes' theorem is a way to find a probability when we know certain other probabilities.
The formula is:
P(A|B) = P(B|A) * P(A) / P(B)
Where:
- P(A|B) is the probability of event A happening given that event B is true.
- P(B|A) is the probability of event B happening given that event A is true.
- P(A) and P(B) are the probabilities of events A and B respectively.
In this case, we want to find P(Caught|Brown), the probability of a rat being caught given that it is brown. We know that:
- P(Brown|Caught) = 0.27, the probability of a rat being brown given that it is caught.
- P(Caught) = 0.37, the probability of any rat being caught.
- P(Brown) = 0.58, the probability of a randomly chosen rat being brown.
Substituting these into the formula gives:
P(Caught|Brown) = P(Brown|Caught) * P(Caught) / P(Brown) = 0.27 * 0.37 / 0.58 = 0.0999 / 0.58 = 0.1724 (rounded to four decimal places)
So, the probability of a rat being caught in a trap given that it is brown is approximately 0.1724 or 17.24%.
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