**1.**Joe, at this point I'd advice students to draw a decision tree. Some would draw one with six nodes in the first layer, representing the machines $M_1, M_2, ... , M_6$ and then 36 nodes in the second layer, representing each of the possible outcomes from $1,2,...,6$ for each of the machines. At each of the branches, they put the possibility to get from one node to the next, and at the end of the the diagram they write down the 36 probabilities for the outcomes which they get by multiplying the probabilities on the branches which lead to the outcome. However, others would opt for a much easier design, summarizing the machines $M_2, M_3, ..., M_6$ as $\overline{M_1}$, and the non-desirable outcomes $\{1,2, ...,5\}$ as $\overline{6}$, which leads to the following graph:

## Sunday, September 2, 2012

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