Determining whether two events are independent or dependent.

Question

I'm trying to make sure that my reasoning is correct for these problems.

Say if the following pairs of events should be modeled as independent or dependent. Explain your reasoning.

We choose a voter at random (all voters equally likely) from Bloomington and let A be the event that the voter votes to reelect the mayor and B be the event that the voter votes to reelect the police chief. (these are not mutually exclusive choices, . . . , a person could vote to reelect both, neither, etc.)

• Independent, because the first person voting on the mayor doesn't affect how the 2nd person votes on the police chief.

Two people are selected at random from Bloomington and let A be the event that the first person favors the mayor, while B is the event that the 2nd person favors the mayor.

• These two events should be modeled as independent because the people were picked at random as well as they their decisions don't affect the other's.

Flip a coin and let A be the event that the coin is heads and B be the event that the coin is tails.

• This event is independent because regardless of how many flip the coin or if you don't do the first coin flip the probability will always be 50%

A person is selected at random from Bloomington. A is the event that the person likes the movie “The Incredibles” while B is the event that the person likes “The Incredibles 2.”

• These variables share a dependent relationship due to the fact that the two items are closely related and if you liked the first one it changes how much you like the second.

Answer 1

• only one voter is selected (why do you speak of first person and second person here?). Police chief and mayor could be one the same line on many aspects. So dependence.
• Indeed independent if you neglect that they have a common background (both come from Bloomington).
• Extreme dependence: the events are even mutually exclusive.
• Dependence.

Answer 2

These are all dependent events. However, the question asks how they should be modelled, which is not (quite) the same thing.

The fourth one is obviously dependent: what a person thinks about the first film is going to be strongly correlated with what they think about the second film. The same applies, less obviously, to the first; we don't know what the relationship might be between a person's view on the police chief and the mayor (depending on local politics, it might be positively or negatively correlated) but there undoubtedly is one.

The second one has a very small amount of dependence, and should be modelled as independent. The dependence comes from the fact that the second person is different to the first, so slightly more likely to have the opposite opinion. In a large population, this dependence is negligible.

The third one I think you have misunderstood the situation. The coin is only flipped once - A occurs if and only if B doesn't, so A and B are dependent. If there were two separate flips then they would be independent.

Answer 3

Applying probability theory to the real world is always problematic, as there may be many external factors not mentioned in the brief problem statement. Phrased differently, you really can't "determine" abstractly whether real world events are independent or not. Lots of things are connected in ways that aren't immediately apparent (famously "ice cream consumption" is positively correlated to "homicide rates" for instance). To be sure, you can make assumptions but in that case the assumption should be clearly labeled as such. Or you can analyze the data and see statistically whether or not it supports the hypothesis of independence.

For your first case, for instance: Perhaps some voters always vote to reelect incumbents (that seems to be true, actually). In that case, knowing that your voter chose to support one incumbent would be evidence that they belonged to this cohort and thus evidence that they'd support all the incumbents. Or, perhaps, some cohort of voters always votes by party line. Or perhaps there is an external factor (such as some political ideology) which induces people to support either both or neither.

Similar for your second example. Perhaps there is some external factor compelling votes in a certain way, perhaps, say, the phrasing of the poll question encourages one vote over another. (Note: this is the only one of your examples where I'd say that assuming independence was at least fairly reasonable).

The third is the very opposite of independent as $A=B^c$.

I think your analysis of the fourth is ok, though I'd phrase it differently. It's not that enjoying part I "changes how much you like" the other. Rather, it is natural to imagine that knowing that you enjoyed part $1$ is evidence for the assumption that you enjoy certain types of film and, as part II is a very similar movie, that becomes evidence that you will enjoy part II. To be sure, it does not have to play out this way. Perhaps we have observed that nobody who liked part I also liked part II (there are probably example of that). In that case we might assume that Part II changed critical elements in such a way as to alienate the earlier fans. Then the evidence issue works in reverse.