The Pelayo Family Plays Roulette

The Pelayo family has been studying roulette wheels and collecting data on tens of thousands of spins. Now the Pelayos have identified a wheel that they want to bet on, which has six numbers that have a 1/30 probability of winning instead of 1/37. They have staked EUR1,000 on this wheel. The question is: How much should they bet in order to have the most capital after 200 rounds? Log utility is used to compare alternatives and find an optimal bet. Excerpt UVA-QA-0850 Rev. Nov. 28, 2016 The Pelayo Family Plays Roulette “Table 17 has the roulette wheel we'll play. Maybe it's bent!” Gonzalo Garcia-Pelayo, the patriarch of the Pelayo family, announced. The Pelayo clan, marking tens of thousands of spin numbers for six hours a night over the course of two weeks, had identified this table at the Casino Gran Madrid (Figure1) as their best opportunity. It appeared that table 17 had six favorable numbers. “Three numbers on each side have a probability of 1/30,” Ivan, one of Gonzalo's sons, said. “Let's play with EUR1,000 for 200 rounds and put equal amounts of money on those six numbers when we play.” Ivan was excited to get started. “Well, we want to avoid the risk of losing the entire EUR1,000 and to have as much as possible at the end of the 200 rounds,” Gonzalo responded. “I calculated that if we bet EUR1 on each of the six favored slots, we get an expected payout gain of EUR0.20 for each EUR1 bet. That's not bad for a few seconds of play. The work will be to decide how much to bet on a round. Based on the minimum bet, we have to stop if our stake gets below EUR6.” . . .