Why systematic random sample for exit polling? I was sitting in class today and we were going over how systematic random samples are used in election exit polling
e.g.
say we want to sample 1/15 people that come to vote. choose some n between 1 and 15 randomly
sample the nth person, and then every 15th person after that.
why couldn't you just assign probability of 1/15 to every person leaving the poll and sample those who end up being chosen to sample?
 A: You can indeed employ a simple random sample.  However, the systematic sample has two slight advantages over the simple random sample.


*

*First, it's probably a little easier and more reliable in the field to enumerate every voter and poll them at regular intervals.  At my polling place, for instance, everybody is sequentially (and publicly) numbered: consequently, there's no mistaking whom to include in the exit poll.  To manage a simple random sample you have to manage a list of random numbers in real time without making mistakes.  Moreover, there are likely going to be clusters of successive numbers in that list, requiring you to poll groups of two or three people, many of whom will not want to wait, causing you to miss their responses.

*Second, the systematic sample is a little more efficient.  The reason is that some people (such as married couples) will arrive and leave in groups and, within those groups, are likely to give similar responses.  The simple random sample will include multiple people within some groups, whereas the systematic sample won't (unless the group is greater than 15 people).
Whether this second effect matters is an interesting question, so I constructed a simple (but realistic) model and simulated the results.  This model creates group sizes with a Poisson distribution (to which 1 is added to eliminate any possibility of a group with nobody in it!).  It supposes the poll asks a single yes/no question and everybody within a group will answer that question in the same way.  (This is a bit extreme, because in reality groups will not be quite so homogeneous in their responses.  But it nicely captures the phenomenon we are concerned about.)  Otherwise, the groups are independent and their chance of answering "yes" is always the same value (equal or close to $1/2$).
I simulated polling places where 1,000 voters appeared (which is typical where I live).  Each simulation created 5,000 independent elections.  For the sample proportion of $1/15$ and average group sizes of $2,$ the variance of the average "yes" proportions among the 5,000 elections was $0.00430$ in the simple random sample but only $0.00374$ in the systematic sample.  (Theoretically, the variance in the simple random sample should equal $1/2(1-1/2)$ divided by $1000/15,$ equal to $0.00375:$ the simulation reproduced that value accurately.)  The ratio of these variances is $1.15.$  This means that the "information" obtained by a systematic sample, expressed in terms of how precisely it estimates the average poll response, is about the same as taking a simple sample of $15\%$ more people.  In this specific sense, the systematic sampling is more efficient.
These numbers of course depend on the particulars.  For instance, with an average group size of $4/3,$ the systematic sample was $14\%$ more efficient.  With an average group size of $4/3$ but polling every third person (rather than every 15th), the systematic sample was $36\%$ more efficient.
Repetitions of the same experiments produced slightly different numbers, but the results were qualitatively the same.  As a control I ran simulations with no grouping.  Returning to sampling every 15th person, the systematic sampling was $5\%,$ $1\%,$ $1\%,$ and $-1\%$ more efficient--that is, only negligibly different from the simple random sampling procedure (as it should be in the absence of grouping).

These simulation results bear out the intuition that for exit polling, one gets a little more for any given unit of effort from a systematic sample compared to a simple random sample.

In some circumstances there is a risk to systematic sampling.  It arises from the possibility of cyclic repetitions of response patterns, which--if it coincides with the sampling frequency--can introduce a systematic bias into the results.  This can happen in systematic samples of geological media, seasonal time series, wave patterns, and so on.  (For example, measuring the temperature of a location every 365 days may give a poor estimate of the average annual temperature, even when the measurements are extended over decades.) Such patterns seem highly implausible in this voting scenario.

For those wishing to explore further--and/or check the correctness of these simulations--here is the R code that performed them.
n.groups <- 1e3 # Election size
lambda <- 1/3   # Average group size is one greater than this
p <- 1/2        # Expected proportion "yes" answers
rho <- 1/15     # Mean sampling frequency
#
# Simulate voting in groups.
#
sim <- replicate(5000, {
  groups <- 1 + rpois(n.groups, lambda)
  y <- cumsum(groups)
  groups <- groups[y <= n.groups]
  n <- length(groups)
  groups[n] <- groups[n] + n.groups - y[n]
  votes <- unlist(mapply(rep, runif(length(groups)) <= p, groups))
  votes
})
#
# Conduct two samples of the elections.
#
sim.srs <- apply(sim, 2, function(x) x[runif(dim(sim)[1]) <= rho])
sim.sys <- apply(sim, 2, function(x) x[seq(1,dim(sim)[1], by=1/rho)])
#
# Report on the variances and their ratio.
#
(a <- var(sapply(sim.srs, mean)))
(b <- var(colMeans(sim.sys)))
(a/b) # Relative efficiency of the systematic sample

