Question

A fox hunts in three territories A, B, and C. He never hunts in the same territory on two successive days. If he hunts in A, then he hunts in C the next day. If he hunts in B or C, he is twice as likely to hunt in A the next day as in the other territory.

What proportion of his time does he spend in A, in B, and in C?

Answer 1

The fox spends 2/3 of his time in territory A, -1/3 of his time in territory B, and 2/3 of his time in territory C.

Step-by-step explanation:

To find the proportion of time the fox spends in each territory, we need to analyze the given information and calculate the probabilities. Let's assign the probabilities as follows:

P(hunt in A) = x

P(hunt in B) = y

P(hunt in C) = z

According to the given information:

1. If the fox hunts in A, then he hunts in C the next day. Therefore, we have the equation x = z.
2. If the fox hunts in B or C, he is twice as likely to hunt in A the next day as in the other territory. This can be written as the equation 2(y + z) = x.

From the first equation, we can substitute z for x in the second equation to get 2(y + x) = x. Simplifying this equation, we find that y = -x/2.

To find the proportion of time the fox spends in each territory, we need to set up the equation x + y + z = 1 and substitute the values we have found:

x + (-x/2) + x = 1

Simplifying, we get 1.5x = 1, which results in x = 2/3, y = -1/3, and z = 2/3.

Therefore, the proportions of time the fox spends in territories A, B, and C are 2/3, -1/3, and 2/3, respectively.