The water continues to rise Your socks are already soaked Yo

The water continues to rise. Your socks are already soaked. You do not have much time left You look around the square room that you have found yourself trapped in and you see a button just to the right of a door in the wall marked, \"Unlock The Door\". You immediately try opening the door. Sure enough, it is locked. You press the button. Immediately a loud and resonant voice thunders slowly and melodically in the confined space: \"Two Suffices\", and a small panel opens on the opposite side of the room revealing a line of 10 small switches, each with three positions, labeled Down Center and Up (D,C,U). All are currently Down. You try the door and it remains locked. You instantly understand that \"Two Suffices\" means that you only need to get two of the ten switches to be set correctly for the water to stop flowing in and the door to unlock. But which two switches and which settings of each? You make a quick estimate of the amount time you have left before the rising reaches your and drowns It might enough water across be for maybe a couple of dozen round trips the room to set the switches and then return to press the button and try the door, assuming you don\'t slip and fall in the rising water. (a) Find some sequence of switch settings that guarantee your survival. E how you developed this sequence and why it works. (b) Estimate the number of round trips across the room you would have to make in order to guarantee your survival adopt a straightforward brute-force approach and the water wasn\'t a factor.

Solution

Out of the 10 switches, a minimum of 2 switches should be set correctly. To find a sequence that guarantees that atleast 2 out of the 10 switches are in their correct position. Let D denote Down, C Center and U Up positions of switches. Total number of switch configurations is 3¹⁰, each has a probabiity of 1/3.

Initially, all the switches are DDDDDDDDDD and the door remains closed. That means this configuration does not have a minimum of 2 switches set to the correct position. At max, only 1 might be in the correct position, i.e., 9 are in their wrong position. Therefore, total number of switch configurations (after conditioning) in the sample space becomes 2⁹. Each of the 9 switches has a probability of 1/2.

Trip 1: You turn all switches to CCCCCCCCCC and return to check the door. If the door does not open, it means that at max., 1 switch is in the wrong position. Therefore, considering the previous case as well, at max., only 2 switches are in their wrong setting.

Trip 2: You turn all switches to UUUUUUUUUU and return to check the door. The door is guaranteed to open, as out of the 10, 8 will definitely be in the correct position.

(b) Total number of round trips across the room in order to guarantee your survival, if we adopt a brute-force approach, i.e., total number of (3¹⁰-1) trips.