4. (5 points) A miner is trapped in a mine and has a choice of three doors. Thou

Question

4. (5 points) A miner is trapped in a mine and has a choice of three doors. Though he doess' realize it. . if he chooses to exit the first door, he is expected to reach safety after 2 hours of travel . if he chooses the second one, he is expected to be back to the mine after 3 hours of travel; and if he chooses the third one. he is expected to be back to the mine after 5 hours of travel. Suppose the doors look identical (thus have equal chances to be chosen) and if he returns to the mine he does not remember which door he tried earlier. What is the expected time until he reaches safety?

Explanation / Answer

Let X denote the random variable that represents the time taken by the miner to reach safety. X {2, · · }. We want to know E[X].

From the given information, E[X | Y = 1] = 2, E[X | Y = 2] = 3 + E[X] and E[X | Y = 3] = 5 + E[X].

Hence, E[X] = E[E[X | Y ]] = E[X | Y = 1] x Pr[Y = 1] + E[X | Y = 2] x Pr[Y = 2] + E[X | Y = 3] x Pr[Y = 3] = 2 x (1/ 3) + (3 + E[X]) x (1/ 3) + (5 + E[X]) x (1/3) = 10 / 3 + (2/3) x E[X]

So, (1/3) E[X] = (10/3) and therefore, E[X] = 10.

The miner can expect to spend ten hours in the mine, before he gets to safety

Related Questions

Navigate

• Browse (All)
• Browse 4
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.