Pinocchio is in his room with the door closed. He didnt come back home from school on time, as he went off to the record store without permission. Geppetto is outside the room asking why he was late. Pinocchio has 4 different lies, lie 1, lie 2, lie 3, and lie 4, that he tells Geppetto with probabilities 0.4, 0.3, 0.2 and 0.1 respectively. Pinocchio is stressed and forgetful so Pinocchio can use the same lie multiple times. Geppetto will believe lie 3, but Pinocchio does not know that. Geppetto asks him why he was late, and Pinocchio tells Geppetto one of the 4 lies, and his nose grows, 1, 2, 3 or 4 inches respectively, regardless of whether Geppetto believes Pinocchio or not. In the case that Geppetto believes his lie, Pinocchio can leave to go to his friends house. In the case that Geppetto does not believe the lie, Pinocchio has to start again and tell him one of the 4 lies, and his nose grows depending on which lie it is, and again, if Geppetto believes him, he can leave and if not he has to tell him one of the 4 lies and his nose grows and on and on till Geppetto believes him. So this is how it works... Step 1) Pinocchio tells a lie based on the lie probabilities Step 2) his nose grows depending on which lie it is Step 3) Geppetto believes him or not. 4) If Geppetto believes him, he can leave, and if Geppetto does not believe him, he goes back to step 1. What is the expected increase in length of Pinocchios nose when he is allowed to go to his friends house?