COU 2: Diagramming Conditional Probability
Imagine a group of persons who must make a collective decision by voting. Specifically, the group must choose between two alternatives, called A1 and A2. Each person in the group will cast a vote for either A1 or A2. If k or more members of the group vote for A1, A1 is adopted, where k is a number between 1 and the total number of persons in the group. If few than k members of the group vote for A1, then A2 is adopted.
For instance, this could be a group of five people deciding between two alternatives by majority rule. In this case, the k would be equal to three, since if three or more of the five persons vote for A1, then a majority have voted for A1, and if fewer than three of the five persons vote for A1, then a majority have voted for A2.
Another example is a jury in an American criminal trial. Juries in criminal trials are made up of 12 persons. Each person casts a vote indicating whether the defendant ought to be deemed “guilty” or “not guilty”. In criminal trials, a defendant can only be found guilty by unanimous decision of the jury. So, k=12 – i.e. if all twelve of the jurors vote “guilty”, the verdict is “guilty”, and if fewer than 12 of the jurors vote “guilty”, the verdict is “not guilty”.
Persons in groups making decisions like these sometimes feel that the ways their peers will vote are informative about which of the available decisions is the right one. For instance, suppose I am a member of a jury and am unsure, given the arguments and evidence I’ve seen at trial, about whether the verdict ought to be ‘guilty’ or ‘not guilty’. But after deliberating with my fellow jurors, I’ve come to respect their judgement. I might then think something like, “well, if all eleven other jurors think that ‘guilty’ is the right verdict, it probably is. So if they will all vote ‘guilty’, so will I”.
We can model this kind of judgement – in which one’s belief about which alternative is better depends on how many of one’s fellow group members would vote for each one – using a probability distribution.
Imagine a member of such a group named Amir. Suppose that Amir is unsure whether alternative A1 or alternative A2 is the better of the two alternatives. Specifically, Amir’s belief about this is given by:
| Event | Probability |
|---|---|
| A1 is the better alternative | \frac{1}{3} |
| A2 is the better alternative | \frac{2}{3} |
Further, Amir thinks that the other members have some ability to discern which alternative is better. So he thinks that the number of his fellow group members who think A1 is the better of the two alternatives depends on whether A1 actually is the better alternative. Specifically, he believes:
| Outcome | Probability |
|---|---|
| Fewer than k-1 of Amir’s fellow group members think A1 is better | \frac{1}{16} |
| Exactly k-1 of Amir’s fellow group members think A1 is better | \frac{3}{16} |
| More than k-1 or more of Amir’s fellow group members think A1 is better | \frac{3}{4} |
| Outcome | Probability |
|---|---|
| Fewer than k-1 of Amir’s fellow group members think A1 is better | \frac{3}{4} |
| Exactly k-1 of Amir’s fellow group members think A1 is better | \frac{1}{16} |
| More than k-1 or more of Amir’s fellow group members think A1 is better | \frac{3}{16} |
Remember, under the assumptions we’ve made above, the group chooses alternative A1 if and only if k or more group members vote for A1. So the way we’ve broken down the possibilities of how many of Amir’s peers think A1 is the better alternative is significant. If fewer than k-1 or more than k-1 of Amir’s peers think A1 is the better alternative and vote accordingly, Amir’s vote has no effect on which alternative is chosen. Amir’s vote is only determinative of the outcome when exactly k-1 of his peers vote for A1, and so the likelihood with which that occurs is important.
Draw a conditional probability diagram depicting the conditional probability distribution above.
Rubric:
You get four points on this COU if the diagram you draw includes all the required information and correctly uses all the relevant graphical conventions.
You get two points if you draw a square with the axes labeled properly and with the space within the square divided up into the appropriate number of boxes, but the relative sizes of any of the boxes are obviously wrong and/or any of the numerical labels for the probabilities are missing or wrong.
You get zero points otherwise.