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I was playing Emerald. Wally was doing the tutorial and, after the Tackle the Ralts fainted. Why did it happen? Will Wally still have the Ralts? Will there be a change in the story or Wally's final team because of this?

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Wille? Did Zigzaggn get a crit?
Not a critical

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In the Generation III games, it is possible for the capture tutorial to end with Ralts fainting. This can only happen if Zigzagoon is generated with 11 Attack, and the wild Ralts has a Nature that lowers Defense, an IV of 3 or less in HP, and 5 or less in Defense. With all this, Zigzagoon can do a maximum of 10 damage with its first Tackle, and after Growl is used on it, 7 with its second. This makes for a total of 17 damage, just enough to knock out Ralts. However, the game continues on as if it had been caught. The probability of this occurring is (182/800)(4/25)(4/32)(6/32)(5/16)*(7/16), or about 1 in 8574.

There is a very, very small chance for the Ralts to be knocked out in the catching tutorial.

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Will willie still have the ralts.
Damn, more rare than finding a random shiny... Didn't even know this could happen. Neat.
"However, the game continues on as if it had been caught." Yes.