Discussion 1:
When we roll one die, we have a 1 in 6 probability of getting any particular number on the die.  When we roll a pair of dice, there are 36 different pairs that can be produced, yet only 11 actual distinct values.

Explain how the probability associated with the roll of each individual die in the pair explains the higher variability in the total outcome of the roll of each pair.  How do the concepts of permutations and combinations apply to this example? Discuss how the notion of degree of freedom can be used to illustrate the accumulating results of a set of dice rolls.

Discussion 2:

Bayes Theorem deals with the calculation of posterior probabilities, which isnt always a natural thing to do. Were used to forward-chaining our probabilities (e.g., if we roll a 3 on a die, whats the probability the second roll will give us a total of 8?).  Backward-chaining is less intuitive (e.g. if our total on the die was an 8, whats the probability that the first roll was a 3?).  Since the rules of probability involve simple addition and multiplication, they work fine in both directions.  The thing that makes posterior probability more difficult is that we simply arent used to thinking about things that way.

Our chapter reading provides an example of a diagnostic test for a rare disease.  The resulting confidence in a positive test result is surprisingly low.  Discuss why that is so.  What is happening in the interaction of the various probabilities that leads to this outcome?

Discussion 3:
The Monty Hall Problem in one of this week’s readings (in Wheelan) offers a perspective on why it is so important to understand and trust probability theory. It’s conclusion is counter-intuitive to some people, so people will sometimes strongly resist changing their minds to obtain a better result. It illustrates how we tend to commit to an opinion or a choice we’ve made even when we should be willing to change our minds in the face of new data.

Explain whether you agree or disagree with the idea of switching as described in the readings, and offer a reason why your opinion should be correct. (Note: Switching is a good idea, so if you disagree you’d better offer a good reason.) Discuss how this scenario might be seen in other decisions that we have to make in engineering, or that we might observe being made by management. Do you think probability theory enters into people’s thinking sufficiently? 