In Search of a Fair Bet in the Lottery

Although state-operated lotto games have the worst average expected payoffs among common games of chance, because the jackpot can accumulate, the maximum expected payoff is potentially unlimited. It is possible, therefore, that lotto can exhibit a positive expected return. This paper examines 18,000 drawings in 34 American lotteries and finds approximately 1 percent of these drawings provided players with a fair bet. If it were possible for a bettor to purchase every possible combination, however, most lotteries commonly experience circumstances where such a purchase would provide a positive return with 11 percent of the drawings providing a fair bet to the player.

Amplified Sensitivity of Nitrogen-Vacancy Spins in Nanodiamonds using All-Optical Charge Readout

Nanodiamonds containing nitrogen-vacancy (NV) centers offer a versatile platform for sensing applications spanning from nanomagnetism to in-vivo monitoring of cellular processes. In many cases, however, weak optical signals and poor contrast demand long acquisition times that prevent the measurement of environmental dynamics. Here, we demonstrate the ability to perform fast, high-contrast optical measurements of charge distributions in ensembles of NV centers in nanodiamonds and use the technique to improve the spin readout signal-to-noise ratio through spin-to-charge conversion. A study of 38 nanodiamonds, each hosting 10-15 NV centers with an average diameter of 40 nm, uncovers complex, multiple-timescale dynamics due to radiative and non-radiative ionization and recombination processes. Nonetheless, the nanodiamonds universally exhibit charge-dependent photoluminescence contrasts and the potential for enhanced spin readout using spin-to-charge conversion. We use the technique to speed up a $T_1$ relaxometry measurement by a factor of five.Comment: 13 pages, 14 figure