## What’s the largest amount won playing roulette?

‘The Man Who Broke the Bank at Monte Carlo’ was a popular British music hall song in the late nineteenth century but, by that stage, half a dozen players had, quite literally, set the bells ringing in the Casino de Monte-Carlo by winning more than the 100,000 franc cash reserve – otherwise known as the ‘bank’ – set aside to cover liabilities on each roulette.

The first man to famously do so was Joseph Jagger, a mechanically-minded, but down-on-his-luck, piece worker from Yorkshire in the North of England. In 1873, working in cahoots with unscrupulous casino staff, Jagger recorded the results of every spin of each roulette wheel at the Casino de Monte-Carlo for a period of weeks. Subsequently, having discovered that one of the wheels displayed a distinct bias towards some numbers rather than others, he began to bet on the frequently-occurring numbers. Over a period of several days, he won 2,000,000 francs, or the equivalent of £7.5 million in modern terms.

Less than two decades later, in 1891, Charles Wells, a known petty criminal born in Hertfordshire in the East of England, but educated at Clermont-Ferrand University in central France, broke the bank at Monte Carlo not once, but several times. Surprisingly, perhaps, given his dubious background, each time he did so by pure good fortune, without resorting to any form of skullduggery, subterfuge, or out-and-out cheating. One the first occasion, when he won 1,000,000 francs, he reportedly won twenty-three of thirty consecutive spins of the roulette wheel and, on his return to Monte Carlo later the same year, he managed to win another 1,000,000 francs from a series of random bets.

## What are the odds against making four-of-a-kind in Texas hold’em poker?

Texas hold’em poker is the most popular ‘community card’ variant of the game, featuring two cards, known as ‘hole cards’, dealt to each player and five more on the board. In other words, each player has the choice of seven cards from which to build the best five-card hand. Of course, a four-of-a-kind hand must include four cards of the same rank – from one of the thirteen ranks available – along with a single card, or ‘singleton’, of some other rank.

Overall, the probability of four-of-a-kind in seven cards is 0.168%, which represents odds of 594/1. In other words, a Texas hold’em player can expect to make four-of-a-kind, or ‘quads’, once every six hundred hands or so, on average. However, while four-of-a-kind is not an unbeatable hand – it ranks behind a straight flush of any description and, in the case of kings or lower, behind four-of-a-kind of higher rank – it is worth waiting for. Indeed, the probability of losing a hand with four-of-a-kind is 0.00001%, or 100,000/1 against.

If a player has paired hole cards, the probability of which is 5.88% or, in terms of odds, 16/1, the probability of ‘flopping’ four-of-a-kind is 0.245% or, in terms of odds, 400/1, although the probability of hitting four-of-a-kind by the fifth, and final, community card, known as the ‘river card’ increases to 2%, representing odds of 50/1. By contrast, a player without paired hole cards has just a 0.001%, or 1,000/1, chance of flopping four-of-a-kind.

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