An eccentric and somewhat absent-minded magician forgot to label his trick coins after performing at a birthday party. The probability of heads for the three trick coins is .30, .40 and .65. He engineered an elaborate contraption to flip the coins and tabulate the results so he could identify which coin was which. He reasoned that the contraption has a constant error rate between anywhere between 0 and .50, such that a head is accidentally recorded as a tail with probability p_error and a tail is accidentally recorded as a head with the same probability. Before putting his contraption into motion, he arbitrarily labelled the coins as a, b, and c.
In this market, I ask two questions. Accordingly, the markets are organized into two sets. The first set of markets ask which coin is which. For example, the market labelled p_a = .30, p_b = .40, p_c = .65 corresponds to the case in which the probability of heads for coin a is .3, the probability of heads for coin b is .40, and the probability of heads for coin c is .65. The second set of markets ask to which interval p_error belongs. For example , the market labeled 0 ≤ p_error < .10 would resolve to true if p_error = .087. The market will be updated once per day according to the schedule in the table below. In total, there will be 7 updates.
Upon request, I will provide a description of the magician's contraption in the comments.
Cumulative heads
N flips Coin A Coin B Coin C
0 0 0 0
5
10
25
50
75
100
200
1,000@kleinquartic here is the lore:
The magician dreaded the time-consuming and tedious task of flipping the coins, and tabulating the outcomes. He had a brilliant idea: he would automate the process with a fantastic machine. After a weekend of feverish research and development, he had engineered the perfect system. It worked like this: He flushed a toilet to trigger a cascade of events, ultimately culminating in a bowling ball falling into a box of iPhones located on one end of a see-saw. This caused the other end of the see-saw to pop up forcefully, thereby flipping the coin. The outcome of the coin flip is adjudicated by an elaborate committee of circus monkeys who worked as independent contractors through the app MonkeyBusiness. The committee is divided into 51 sub-committees, each with at least 3 members. Each sub-committee votes on the outcome by screaming into a sound level meter and the side (heads vs. tails) with the highest decibels determines the recorded result. The results are tallied across sub-committees and the majority result determines the official record. Finally, an appointed monkey ceremoniously presses the button corresponding to the official result, which causes a machine to emit a pleasant bell tone and dispense a piece of funfettie cake.
The magician pulled the lever on the toilet and watched his creation spring into life. A cacophony of monkey shrieks, intermittent thumps, and mechanical sounds filled his modest bungalow. As he stood there gloating upon his hard work, a nagging feeling of doubt began to build. Could this system be prone to error?