2) Capture-Mark-Recapture/Resight (CMR) is a common method for determining survival in animal populations. The Cormack-Jolly-Seber approach can be used to estimate survival from CMR data, resulting in two parameters being estimated - the apparent survival (, phi) or recapture/resight probability (p). Recapture probability is obviously conditional on the animal surviving (i.e., dead animals can't be recaptured). Animals are recaptured at regular intervals (e.g., yearly sampling periods). Data are stored in capture histories in which the column indicates the first sampling period (when animals are first marked) and 1's indicate capture/recapture, and 0's indicate lack of capture. In a simple 2 sample period case, this results in the following possibilities:

With 2 occasions there are 2 possible capture histories (11, 10) created by 3 possible scenarios (lives, seen; lives, not seen; dies). The probability of each scenario is p, (1-p), and 1-, respectively, although we can't separate the last 2 scenarios with only 2 capture periods. Based on this:

a) What is the event space for a 3 occasion study? Provide the capture histories and required scenario for each. Base your event space only on the possible fates of those individuals captured on the first occasion (e.g., the 3 possible scenarios in the example above do not include a 01 or 00 capture history).

b) Assuming that apparent survival and capture probability do not change from year to year and are .85 and .42, respectively. What is the probability of each capture history being observed?