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

Answer:

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.