Assuming this was completely unaffected by serene grace or moves that can thaw Pokemon or anything like that, ice beam has a 10% chance of freezing on the first try.

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Every time a Pokemon attempts to move while frozen, it has an 80% chance to remain frozen.

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If it attempts to move twice, then it has a 20% chance to thaw on the first try, so it has an 80% chance to be frozen on the second turn. Then, it has another 80% chance to stay frozen again, so the chance of staying frozen for 2 turns is 80% times 80%, or 80%^2, or 64%. If you repeat this pattern 6 times and put the 10% chance at the beginning, then you get 10%*80%^6, which is around 1/38.