The story of David and Goliath is probably the biggest underdog upset in the Bible. In modern sports history, however, underdog wins are still rare enough to be considered something of a miracle. Mathematical experts will tell you that according to bookmakers’ odds, punters undervalue the small returns they can make on betting the favourite and overvalue the big returns offered on long-shot bets, even though the former are much more likely to deliver a win.

That is because occasional big windfalls are much more fun than cautious, statistically correct gaming. If you stick to small, affordable bets, you will have a lot better time picking the outsiders with attractive odds than you will trying to chase favourites for more modest, but more probable returns.

Our bias towards winning helps this effect, of course. It is a lot easier to remember long shots that surprisingly came off, than it is to list all the underdogs that delivered exactly the result that the bookies predicted. Here are four of the most memorable upset wins:

St Louis Cardinals V Atlanta Braves

Stuck 4.5 games behind the Atlanta Braves in the 2011 baseball season, with 15 games left to play, the St Louis Cardinals weren’t rated by many. They were being offered at odds of 999/1 to win the National League pennant and go on to win the World Series. Punters who took a chance on them were well rewarded: The Cardinals came back with 11 wins in 15 games to take the pennant, eventually beating the Texas Rangers to win the World Series.

US Hockey Team V Soviet Union

At the height of the Cold War in 1980, the Soviet Union had a formidable professional ice hockey team, sent to the Winter Olympics in Lake Placid, USA with the intention of taking the sporting and political bragging rights. The US squad, in contrast, was made up of amateurs and college players who were not expected to get into the medals; they were priced at 1,000/1 when they took on the Soviet team for gold. What followed was the “miracle on ice”, and a 4-3 US victory.

Buster Douglas V Mike Tyson

Tyson was undefeated when he fought Douglas, having won 33 of his 39 fights by knockouts. Douglas, with four losses on his record, was seven years older than Tyson when they met in Japan in February 1990. Bookies and fans were stunned when Douglas, priced at 42/1, knocked Tyson out in the tenth round.

Leicester City and the Premier League

In 2015, British football club Leicester City narrowly escape relegation out of the Premier League, not for the first time. They were not expected to pose any threat to stronger clubs like Arsenal, Manchester United or Chelsea in the upcoming season, and bookies rated their chances of taking the 2016 Premier League title at 5,000/1 – stupendous odds. The powerhouse teams were all having an off year, it turned out. An unstoppable Leicester launched an unexpected run of victories that confounded the odds and brought home the trophy for the first time in the club’s history.