9. A lottery has a winner every day. The winning number is chosen randomly each day, and the probability of winning on any one day is 0.03. Alice plans to buy a lottery ticket every day for 500 days, and Bob plans to buy a ticket every day for 5,000 days. Which of the following is the best assessment of how the law of large numbers applies to this situation? (A) Alice will probably have a higher propor- tion of wins than Bob. (B) Bob will probably have a higher propor- tion of wins than Alice. (C) The difference between Alice's propor-

Q. 1 It is estimated that 80% of emails are spam emails. Some software has been applied to filter these spam emails before they reach our inbox. A certain brand of software claims that it can detect 99% of spam emails, and the probability for a false positive (a non-spam email detected as spam) is 5%. Now if an email is detected as spam, then what is the probability that it is in fact a non-spam email? event A: email is spam; event B: email is detected as spam. Assume that

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Example 7.40: You set out on a walk with three possible locations, denoted locations 1, 2 and 3. The probabilities of travelling from a specified location to the next are given by Solution. If you know that you started in location 3, what is the probability that you end up at location 1 after two time periods? Using the above equation, we get To solve this we use the equation Xn+1 = AXn. Given that you began in location 3, the initial state vector is and subsequently Exercise X₁ = AX0 Probability = A = = X₂ = AX₁ 0.5 0.2 0.4 0.4 0.6 0.1 0.1 0.2 0.5 = Xo 0 0.5 0.2 0.4 0.4 0.6 0.1 0.2 0.5 0.1 ][ Therefore the probability of ending up in location 1 after two time periods is 0.42. 0.5 0.2 0.4 0.4 0.6 0.1 0.1 0.2 0.5 1 0.4 0.1 = 0.5 0.4 0.1 0.5 = 0.42 0.27 0.31 Refer to the example above. If you know that you started in location 2, what is the probability that you will end up at location 1 after one time period?