Play   now? Play later? You   can become a millionaire! That’s what   the junk mail said. But then there was   the fine print: If   you send in your entry before midnight tonight, then here are your chances: 0.1%   that you win $1,000,000 75%   that you win nothing Otherwise,   you must PAY $1,000 But   wait, there’s more! If you don’t win   the million AND you don’t have to pay on your first attempt, then   you can choose to play one more time. If you choose to play again, then here are your chances: 2% that   you win $100,000 20%   that you win $500 Otherwise,   you must PAY $2,000 What   is your expected outcome for attempting this venture? Solve this problem using a   decision tree and clearly show all calculations and the expected monetary   value at each node. Use   maximization of expected value as your decision criterion. Answer   these questions: 1)   Should you play at all? (5%) If you   play, what is your expected (net) monetary value? 2) If   you play and don’t win at all on the first try (but don’t lose money), should   you try again? Why? 3)   Clearly show the decision tree (40%) and expected net monetary value at each   node

1) To determine whether you should play at all, we need to calculate the expected net monetary value. In a decision tree, we start with the initial node and calculate the expected value at each subsequent node.

Let’s begin with the initial node, where you can choose to play or not to play. If you choose to play, there is a 0.1% chance of winning $1,000,000 and a 75% chance of winning nothing. The remaining probability of 24.9% corresponds to the chance that you must pay $1,000. Therefore, the expected value at this node can be calculated as follows:

Expected value = (0.001 * $1,000,000) + (0.75 * $0) + (0.249 * (-$1,000))
Expected value = $1,000 + $0 + (-$249)
Expected value = $751

So, if you decide to play, the expected net monetary value on the first try is $751.

2) Let’s consider the possibility of playing again after not winning on the first try. At this node, there is a 2% chance of winning $100,000, a 20% chance of winning $500, and an 78% chance of paying $2,000. We can calculate the expected value at this node:

Expected value = (0.02 * $100,000) + (0.20 * $500) + (0.78 * (-$2,000))
Expected value = $2,000 + $100 + (-$1,560)
Expected value = $540

So, if you decide to play again, the expected net monetary value on the second try is $540.

3) Now, let’s construct the decision tree with the calculated expected values at each node.

[Decision Node]
/
[Play] [Don’t Play]
/
[Win $1,000,000] [Win Nothing/ Pay $1,000]

/
[Play Again] [Don’t Play Again]
/
[Win $100,000] [Win $500 / Pay $2,000]

Now we can determine the expected net monetary value at each node:

– Expected value at [Win $1,000,000] node = $1,000,000
– Expected value at [Win Nothing/ Pay $1,000] node = -$1,000
– Expected value at [Play Again] node = $540
– Expected value at [Win $100,000] node = $100,000
– Expected value at [Win $500 / Pay $2,000] node = -$1,500

From the decision tree, we can see that the expected value at the [Play] node is $751, which is higher than the expected value at the [Don’t Play] node, which is 0. Therefore, you should choose to play.

Overall, the expected net monetary value of attempting this venture, considering all possible outcomes, is $751.

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