From the Super Mario Wiki
Badminton (Singles) is a Court & Field Sports Event that appears exclusively in the Nintendo 3DS version of Mario & Sonic at the London 2012 Olympic Games. The event focuses on judging whether a shot will land in or out of the court, and awards points based on whether the player guesses correctly. The starting best record for this event is 35pt(s).
Badminton (Singles) appears in the game's story mode as part of the second challenge of Tricksters Episode 1, Waluigi's Ambition, in which Waluigi must beat Metal Sonic in it, along with Handball and Taekwondo. It also appears as the second challenge of Tricksters Bonus 2, Dr. Eggman's Robots, in which Metal Sonic must beat Waluigi in it. It appears again as the final part of the challenge of Tricksters Bonus 2, An Old Urn, in which Metal Sonic must beat Wario in it alongside 100m Breaststroke and Handball.
In this game, the player must judge whether the shuttlecock coming towards their character will land in or out of the court. If the player thinks that it will land in, they must hit the A Button to hit it, or must press left on the Control Pad if they think that it will land out of the court. The player can press the button before the shuttlecock reaches them, and their decision will be indicated by either a red exclamation mark for hitting it and a lightbulb for dodging it. The player may change their decision before the shuttlecock reaches them if they have enough time. Performing the correct move, whether the shuttlecock is in or out, will award the player one point. If the player fails to press a button, presses too late or chooses the wrong option, they will lose one of their three lives. Losing all three will result in the end of the minigame. Toad acts as a referee in this event.
1) Hit: A Button.