Yes. If you did not use any of the 50 free "shows" before 23:59, it will still be 60 the next day. If you used up all the 50 free "shows" before 23:59, even then it will be 60 the next day.
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The number of available free credits will be displayed, every time you click "show". Also, the number of "shows" of your predictions will also be added to the statistics, and may indirectly indicate the reputation of the predictor.