Question

A hunter has two hunting dogs. One day, on the trail of some animal, the hunter comes to a place where the road diverges into two paths. He knows that each dog, independently of the other, will choose the correct path with probability p. The hunter decides to let each dog choose a path, and if they agree, take that one, and if they disagree, to randomly pick a path (probability for each path is 0.5). Is his strategy better than just letting one of the two dogs decide on a path?

Answer 1

The hunter's strategy of letting both dogs choose a path and randomly picking one in case of disagreement is better than relying on just one dog.

Step-by-step explanation:

The hunter's strategy is better than just letting one of the two dogs decide on a path. To understand why, let's consider the different scenarios:

If both dogs choose correctly (with probability p), the hunter will take the correct path.

If both dogs choose incorrectly (with probability (1-p)), the hunter will randomly pick a path.

If the dogs disagree, there are two cases to consider:

• If the hunter chooses the correct dog's path, the hunter will take the correct path (with probability 0.5).
• If the hunter chooses the incorrect dog's path, the hunter will randomly pick a path (with probability 0.5).

So, in both cases of disagreement, the hunter has a chance of taking the correct path. Therefore, the hunter's strategy is better than relying on just one dog.

Answer 2

No. Both strategies has the same chance of success.

Step-by-step explanation:

The base strategy is let one dog choose the path. It has a chance of success of value
`p`.

The strategy that the hunter uses is going wherever the two dogs go, if they choose the same, or pick a path 50-50, if they don't.

Probability of both dogs choosing the correct path =
`p^(2)`

Probability of both dogs choosing the wrong path =
`(1-p)^(2)`

Probability of both dogs choosing different paths =
`1-(p^(2)+(1-p)^(2))`

The probability of the hunter going the right path is

(Probability of both dogs choosing the correct path) + 0.5 * (Probability of both dogs choosing different paths)

`P=p^(2) + 0.5 *(1-(p^(2)+(1-p)^(2))\\P=p^(2) + 0.5 * (1-p^(2)-(1-p)^(2))\\P=p^(2) +0.5-0.5*p^(2)-0.5*(1-2p+p^(2))\\P=p^(2) +0.5-0.5*p^(2)-0.5+p-0.5*p^(2)\\P = p`

The probability of success of the hunter's strategy is the same as letting one dog decide one path.