Trapped Miner

Probability Level 2

A miner is trapped in a mine containing 3 3 doors.

  • The 1 st 1^\text{st} door leads to a tunnel that will take him to safety after 3 3 hours.
  • The 2 nd 2^\text{nd} door leads to a tunnel that returns him to the mine after 5 5 hours.
  • The 3 rd 3^\text{rd} door leads to a tunnel that returns him to the mine after 7 7 hours.

At all times, he is equally likely to choose any one of the doors. What is the expected time to reach safety (in hours)?


The answer is 15.

Brian Lie by Brian Lie , 26, China

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1 solution

X X
Oct 9, 2018

Let the expected time be E E . Then E = 1 3 × 3 + 1 3 × ( 5 + E ) + 1 3 × ( 7 + E ) E=\dfrac13\times3+\dfrac13\times(5+E)+\dfrac13\times(7+E) .

3 E = 15 + 2 E , E = 15 3E=15+2E,E=15

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