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Memory Alpha
Multiple realities
(covers information from several alternate timelines)

Luck was superstitious supposed influence on probability, whether a favorable outcome ("good luck") or an unfavorable one ("bad luck"), while those for do well at something on their first attempt were said to have "beginner's luck". Luck was a popular concept in gambling.

Earth traditions involving good luck include the Blarney Stone and to pat the Buddha's belly. (ENT: "The Andorian Incident")

Trip Tucker wished Captain Jonathan Archer good luck in trying to let the Vulcans admit their star charts were not accurate. (ENT: "The Andorian Incident")

In 2258 of the alternate reality, Spock wished his younger counterpart "good luck" instead of his customary farewell, the latter of which he considered would be, in this case, "oddly self-serving." (Star Trek)

Before Spock embarked on a scientific mission into the space amoeba in 2268, Leonard McCoy expressed a lack of understanding of Vulcan dignity, so Spock asked him to employ a Human superstition and wish him luck. (TOS: "The Immunity Syndrome")

Data had Vanessa blow on his dice for good luck. According to the fictional character Texas, it was bad luck to count your chips at the table. (TNG: "The Royale")

Some regarded luck as a lady, "Lady Luck." When a replica of Jean-Luc Picard interrupted a poker game on the USS Enterprise-D, telling Geordi La Forge that he would like to get engine efficiency up to 95%, La Forge stated that he would get right on it. The replica replied that it was not an order and he could get to it later, but La Forge replied that it was no problem and left, stating that "Lady Luck" had left him long ago. (TNG: "Allegiance")

Julian Bashir and Miles O'Brien wished Odo "good luck" before the constable departed to deal with Changeling infiltration on Earth. Odo appreciated the sentiment, though he said that he did not personally believe in luck. (DS9: "Homefront")

Not long after this, when Odo and Major Kira Nerys were having a discussion regarding Quark being disturbed by noise made by Odo's shapeshifting, Kira commented that it was just Quark's luck that Odo would be assigned quarters right above his. Odo replied that luck had nothing to do with it, much to Kira's amusement. Later, when Bajoran First Minister Shakaar Edon was on Deep Space 9, Kira wished him luck prior to him giving a speech. Sometime after this, Worf noted to Odo that Miles O'Brien had a tendency to drop by his quarters announced and suggested that O'Brien might stop if he was less hospitable. Despite his lack of belief in it, Odo wished Worf "good luck." (DS9: "Crossfire")

Senator Letant stated that Romulans did not believe in luck, and Elim Garak stated that Cardassians don't either. Conversely, General Martok expressed belief in luck. (DS9: "Tears of the Prophets", "Distant Voices")

Michael Eddington owned an heirloom coin that he called his "lucky looney". (DS9: "Blaze of Glory")

When Seven of Nine discussed a precaution against theta radiation with Tuvok, he stated, "With any luck, we won't be needing it." Seven replied that she had thought Vulcans didn't believe in luck, but Tuvok replied that serving with Captain Kathryn Janeway had taught him otherwise. Seven agreed that Janeway did seem to succeed more often than random chance would allow for, and stated that she herself would factor it into her own calculations. (VOY: "Juggernaut")

"Beginner's luck" was a type of luck sometimes attributed to those who win a game on their first try, sometimes even against an experienced opponent. In 2378, when Icheb won a game of kal-toh against Tuvok, Icheb attributed the victory to beginner's luck, though Tuvok recognized his loss as being due to lowered neuropeptide levels caused by a Vulcan neurodegenerative disease. (VOY: "Endgame")

A bit later that year, Admiral Kathryn Janeway arrived from the future of an alternate timeline, with a plan to return the USS Voyager to the Alpha Quadrant by utilizing a Borg transwarp hub. Her captain counterpart was wary, stating that they shouldn't push their luck, but the admiral replied that luck wouldn't have anything to do with it, as she had developed weapons and technology to defeat the Borg. (VOY: "Endgame")

In the script of TOS: "Court Martial", Areel Shaw informally wished James T. Kirk "good luck," which Kirk at first responded "[wasn't] very professional," considering Shaw was meanwhile serving as prosecutor in Kirk's court martial.

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