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Part III: Unguided Problems 1. It is believed that there is cheating at 10% of the tables in a certain casino. To determine which tables, the supervisors have singled out a customer (labeled X) involved in the cheating. If there is cheating at a table, then X has an 80% chance to win. If there is no cheating, there is only a 2% chance to win. Now, X wins at the next table. What are the chances there is cheating at that table? 2. There are two coins-one is fair, one has heads on both sides. You choose a coin at random, toss it, and it comes up heads. What is the probability you tossed the two-headed coin? 3. There are two dice-one is fair, and one is "loaded." The loaded die has a 1/4 chance of giving an ace. The remaining sides are equally likely. You choose a die at random, roll it, and get an ace. What are the chances you rolled the loaded die? Recall an "ace" is a one. 4. In this game, you roll a blue die and a red die, then draw a ticket from a box that is either a winning or a losing ticket. Some boxes give you better chances to win than others. If the blue die lands on five, but not the red, then you have a 40% chance to win. If the red die lands on five, but not the blue, then you have a 50% chance to win. If both dice land on five, then you have a 90% chance to win. If neither die lands on five, then you have onlya 2% chance to win. What are the chances that you will win? 5. There is a 5% chance a computer is infected with a certain virus. When a computer is infected, a certain program has a 90% chance to crash. When not infected, the program still has an 8% chance to crash. If the program crashes, what is the probability the computer is infected with the virus? 6. Suppose about 75% of someone's PC game library is available through Steam and about 45% is available through GOG (good old games). Moreover, 20% of their library is available through both. (a) If one of their games is available through Steam, what are the chances it is also available through GOG? (b) If one of their games is available through GOG, what are the chances it is also available through Steam