Sometimes I forget a few items when I leave the house in the morning. For example, here are probabilities that I forget various pieces of footwear: left sock 0.2 right sock 0.1 left shoe 0.1 right shoe 0.3 1. Let X be the number of these that I forget. What is E(X)? 2. Upper bound the probability that I forget one or more items. Make no independence assumptions. 3. Use the Markov Inequality to upper bound the probability that I forget 3 or more items. 4. Now suppose that I forget each item of footwear independently. Use Chebyshev's Inequality to upper bound the probability that I forget two or more items. 5. Use Murphy's Law to lower bound the probability that I forget one or more items.