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6-Approximate Q-Learning and Probability_sol

6-Approximate Q-Learning and Probability_sol - CS 188...

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CS 188 Spring 2011 Introduction to Artificial Intelligence Section 6 Solutions Approximate Q-Learning and Probability Q1. Mini Power Pellets Note: For this problem, any answers that require division can be left written as a fraction. PacLabs has just created a new type of mini power pellet that is small enough for Pacman to carry around with him when he’s running around mazes. Unfortunately, these mini-pellets don’t guarantee that Pacman will win all his fights with ghosts, and they look just like the regular dots Pacman carried around to snack on. Pacman just ate a snack ( P ), which was either a mini-pellet (+ p ), or a regular dot ( - p ), and is about to get into a fight ( W ), which he can win (+ w ) or lose ( - w ). Both these variables are unknown, but fortunately, Pacman is a master of probability. He knows that his bag of snacks has 5 mini-pellets and 15 regular dots. He also knows that if he ate a mini-pellet, he has a 70% chance of winning, but if he ate a regular dot, he only has a 20% chance. (a) What is P (+ w ), the marginal probability that Pacman will win? P (+ w ) = P (+ w, + p ) + P (+ w, - p ) = P (+ w | + p ) P (+ p ) + P (+ w | - p ) P ( - p ) = 7 10 × 1 4 + 2 10 × 3 4 = 13 40 = 0 . 325 (b) Pacman won! Hooray! What is the conditional probability P (+ p | + w ) that the food he ate was a mini-pellet, given that he won?
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