how to create a variable with conditional distribution

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Hi, dear friends,
May I ask for help for the following code:
I want to create a random number X,
  1. original X (unconditional X) follows a normal distribution with mean = 12.21 and SD=0.63.
  2. given Y=11.50, the probability (X>=Y)>=0.97.
I think this is not a simple truncated normal distribution because the probability (X>=Y) is not 100%.
Can any one give me some help?
I really appreciate
Best,
Yanting

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