Question

Sherlock Holmes finds paw prints at the scene of a murder, and thinks that they are either from a dog, with probability 3=/, or from a small bear, with probability 1/4. He then discovers some unusual scratches on a nearby tree. The probability that a dog would produce these scratches is 1/10, while the probability that a bear would is 3/5. What is the probability, given the presence of scratches, that the animal is a bear?

Answer 1

The probability that the animal is a bear given the presence of scratches, we use Bayes' theorem along with the given probabilities of a dog or bear making the scratches and the prior probabilities of the presence of a dog or bear.

Step-by-step explanation:

The problem presented is a case of Bayesian inference and requires the use of conditional probability to find the likelihood that an animal is a bear, given the presence of unusual scratches on a tree.

To calculate the conditional probability that the animal is a bear given the scratches (P(Bear|Scratches)), we need to use Bayes' theorem:

P(Bear|Scratches) = (P(Scratches|Bear) * P(Bear)) / P(Scratches)

Firstly, we calculate the probability of the scratches happening regardless of the animal by considering both possibilities (dog and bear):

• P(Scratches) = (P(Scratches|Dog) * P(Dog)) + (P(Scratches|Bear) * P(Bear))
• P(Scratches) = (1/10 * 3/4) + (3/5 * 1/4)

Then we substitute the values into Bayes' theorem:

• P(Bear|Scratches) = (3/5 * 1/4) / P(Scratches)

After calculating P(Scratches), substitute it into the equation and solve for P(Bear|Scratches).