Random Variables and their PDFs

 Solutions

1. P(Male | undecided) = 16/33 = .485

P(Female | undecided) = 17/33 = .515

So undecideds are about equally divided bewteen males and females.

2. Nope Dependent. P( Female | Yes) = 10/33 = .417

P(Female) = 52/110 = .46

So knowing that someone is a Bush supporter makes them slightly less likely to be female. Of course you have to be careful how you phrase a sentence like that since it does seem to imply a fluidity in gender that most people don't possess.

Note that P(Female|No) = 25/53 = .47 and this is very close to P(Female). This might lead you to conclude they were independent. However, for "gender" and "bush support" to be independent, this equality needs to hold for all values of the variables. It's sufficient to find one for which the equality does not hold to eliminate independence.

3. P(opinion| Yes Bush ) = x/31 and so the most likely opinion is the one with the most votes in this column. Among Bush supporters, you are most likely to find someone in support of gay marriage. (My classroom is, perhaps, not a represntative sample.) P(Yes gay mrg| Yes Bush) = 16/31 = .516

4. You might --- you should -- expect dependence.

P(Yes Gay Mrg | No Bush) = 25/53 = .471

P(Yes Gay Mrg) = 50/110 = .454

I chose this example because they are pretty close, and students often struggle over how close they need to be. I tell them that if we are taking these 110 people to be a sample of the population, then there's some "wiggle" room (and we'll be more precise about how much wiggle room is allowed when we deal with inference.) But if we take them just as 110 students, and we are selecting from within this group, then there's no wiggle room. Still, you might argue that if just one person were to change their minds, they would be nearly equal. But in this case the argument is irrelevant, because for other values of the variable the difference is greater.

P(Yes gay mrg | Yes Bush) = 16/31 = .516

P(Yes gay mrg) = .454

More proof this class is odd --- support for gay marriage is higher among Bush Supporters than non-Bush supporters. I confess that this might be, in part, because of the way "support for gay marriage" was worded -- there were two different versions of the question, and "support for gay marriage" is a bit of a stretch.

5. P(Yes gay mrg | Yes Bush) = 16/31=.516. Yes, you can do it with a tree diagram. I will let you try...