Daniel Leeds, Michael Leeds & Akira Motomura Economic Inquiry, forthcoming
We use playing time in the National Basketball Association to investigate whether sunk costs affect decision making. Behavioral economics implies that teams favor players chosen in the lottery and first round of the draft because of the greater financial and psychic commitment to them. Neoclassical economics implies that only current performance matters. We build on previous work in two ways. First, we better capture potential playing time by accounting for time lost to injuries or suspension. Second, we use regression discontinuity to capture changes when a player's draft position crosses thresholds. We find that teams allocate no more time to highly drafted players.
So, if "THAT" is the chance of the Michigan lottery having exactly the same number two days in a row, then THAT is pretty unlikely. It's 1/10,000 every day, because it's the chance of hitting yesterday's number again today.
But if we are talking about one state lottery somewhere (there was nothing special about it being Michigan, ex ante) on some day in given year (there was nothing special about those two days), then THAT is just the chance one lottery out of 44 picks the same number on consecutive days, out of 365 (since it could happen on the first day, but that would be across years).
If there are 44 4-digit lotteries every day, and the probability of getting a different number in each particular lottery is 9,999/10,000, that means that the probability of duplicate numbers in SOME state (out of 44), on a given day, is .00439.
But we do that 365 times per year. Since the chance of no duplicates in all 44 states, on a given day, is .9956, the chances of no duplicates for a year is .9956^365 or .2007.
If that's right (and I'm just doing this back-of-the-envelope, so I've probably made a mistake in logic or calculation!), that means that in any given year the chances of a duplicate lottery, two consecutive days the same number, in some state, is about 80%.
Does that sound right? If you carry out to multiple years, say 5 years, the chances of getting at least one duplicate in at least one state are better than .999. It will be a little more complicated in real lotteries, because they are not all simple "pick four digits between 0 and 9," but the same sort of logic applies.
With the caveat, again, that I have likely made a mistake. The question, then, is whether consecutive duplicates are really as common as this calculation implies. Thoughts?