Answer:
(i) 
(ii) 
(iii) 
(iv) 
Step-by-step explanation:
Let  be the event that the dog has gone home,
 be the event that the dog has gone home,
 be the event that the dog has gone to the picnic area, and
 be the event that the dog has gone to the picnic area, and 
 be the event that the dog has gone to the park.
 be the event that the dog has gone to the park.
Assuming all the three events are equally likely to happen, so,

Now, let  and
 and  bet the event of found and lost of the dog.
 bet the event of found and lost of the dog.
Given that the chances of finding the dos are  and
 and  , if the dog is in the picnic area and woods respectively.i.e
, if the dog is in the picnic area and woods respectively.i.e
 ,
, 
 and
 and


(i) The probability that the dog will be found in the park
 Probability of going the dog to park
 Probability of going the dog to park  Probability of found in the park
 Probability of found in the park
 [using equations (1) and (2)]
 [using equations (1) and (2)]


(ii) If the dog is in home, the chance of finding the dog is 100%.
So, the probability that the dog will be found at home
 The probability that the dog has gone home
 The probability that the dog has gone home

 [ from equation (1)]
 [ from equation (1)]
(iii) Given that the dog is found in the park, so, the probability of founding the dog in the picnic area of the park

 [using equations (1), (2) and (4)]
 [using equations (1), (2) and (4)]

(iv) Given that the dog is lost, so, the probability of losing the dog in the woods

 [using equations (1), (3) and (5)]
 [using equations (1), (3) and (5)]
 .
 .