Lost in Bandrika
I’ve been busy studing for my quals lately… at the slow rate of about 1 problem per day. In the interest of sharing the joy,
You are lost in the National Park of Bandrika. Tourists comprise two-thirds of the visitors to the park, and give a correct answer to request for directions with probability 3/4. (Answers to repeated questions are independent, even if the question and the person are the same). If you ask a Bandrikan for directions, the answer is always false.
- You ask a passer-by whether the exit from the park is East or West. The answer is East. What is the probability that it is correct?
- You ask the same person again, and receive the same reply. Show the probability that it is correct is 1/2.
- You ask the same person again, and receive the same reply. What is the probability that it is correct?
- You ask for the fourth time, and receive the answer East. Show that the probability it is correct is 27/70.
- Show that, had the fourth answer been West instead, the probability that that East is nevertheless correct is 9/10.
— Grimmet and Stirzaker 3rd ed prob 35 in chapter 1.
Here’s another (more interesting, and easier) probability problem:
10 per cent of the surface of a sphere is coloured blue, the rest is red. Show that, irrespective of the manner in which the colours are distributed, it is possible to inscribe a cube in S with all its vertices red.