Unfortunately, Eddard is lost in the dark dungeon of the Red Keep. He knows ther
ID: 3329417 • Letter: U
Question
Unfortunately, Eddard is lost in the dark dungeon of the Red Keep. He knows there are three doors, and one of the doors will lead him to freedom. If Eddard takes Door A, he will wander around the dungeon for 2 days and return to where he started. If he takes Door B, he will wander around the dungeon for 3 days and return to where he started. If he takes Door C, he will find the exit after 1 day. If Eddard returns to where he started, he immediately picks a door to pass through, and since it's pitch black in the dungeon, Eddard picks each door uniformly at random. How long, on average, will Eddard wander in the dungeon before finding his way out?Explanation / Answer
There is only Door C, where he will find the exit. So, eddard's quest to find exist door will always end with Door C.
so possibilitis of finding Exit.
(i) Door A -> Door B - > Door C
Expected Time taken in this way = 2 + 3 + 1 = 6 days
Probability of that event = Pr(Door A) * Pr(Door B) * Pr( Door C) = 1/3 * 1/2 * 1 = 1/6
(ii) Door A - > Door C
Expected time takenn in this way = 2 + 1 = 3 days
Probability of that event = Pr(Door A) * Pr(Door C) = 1/3 * 1/2 = 1/6
(iii) Door B - > Door C
Expected time taken in this way = 3 + 1 = 4 days
Probability of that event = Pr(Door A) * Pr(Door C) = 1/3 * 1/2 = 1/6
(iv) Door B -> Door A - > Door C
Expected Time taken in this way = 2 + 3 + 1 = 6 days
Probability of that event = Pr(Door B) * Pr(Door A) * Pr( Door C) = 1/3 * 1/2 * 1 = 1/6
(v) Door C
Expected Time taken in this way = 1 day
Probability of that event = Pr(Door C) = 1/3
On Average, Eddard take expected time in dungon = xp(x)
= 1/6 * 6 + 1/6 * 3 + 1/6 * 4 + 1/6 * 6 + 1/3 * 1 = 3.5 days
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