Computational Statistics question I've got a tricky computational statistics problem and I was wondering if anyone could help me solve it.
Okay, so in your left pocket is a penny and in your right pocket is a
dime. On a fair toss, the probability of showing a head is p for the penny and d for the
dime. You randomly chooses a coin to begin, toss it, and report the outcome (heads
or tails) without revealing which coin was tossed. Then you decide whether to use
the same coin for the next toss, or to switch to the other coin. You switch coins with
probability s, and use the same coin with probability (1 - s). The outcome of the
second toss is reported, again not reveling the coin used. 
I have a sequence of heads and tails data based on these flips, so how would I go about estimating p, d, and s?
 A: This is an example of a "Hidden Markov Model."  There's a huge literature on how to estimate that transition probabilities and how to use the observed data to estimate the current state.  In this particular case there's no way to distinguish between the two coins, but you could obtain estimates of the probabilities of heads for one coin and the other from the observations.  
A: Each flip of each coin is independent -- dime or penny, right or left pocket.  If you get heads on a flip of the penny, it's still equally probable that you'll get heads or tails flipping the dime or the penny again. So taken independently, p = 0.5 and d = 0.5.
However, if you're trying to find the probability of a particular sequence of flip results, the probabilities are different, depending on how many flips and how many ways there are to get that particular sequence, but it still makes no difference which coin or pocket.  Due to their independence, you can't find s (unless one of the coins isn't fair).
That said, there are some studies showing interesting coin flip probabilities under certain conditions.  See Wolfram's Coin Tossing page for some of those.
