Per my response above, multiplying the probability of winning a tag by the number of tags doesn’t work. You could easily get a result where your probability is greater than 100%.
For example, let’s say there are 50 prizes (ie. tags), 100 tickets sold (ie. total points in the draw) and you have 10 tickets (points). Your probability of winning the first drawn is 10/100=10%. Using your math, if there’s 50 tags to hand out then you’d say your probability of winning any one of those would be 0.1*50 = 5.0 (or 500%). That can’t be right because there’s definitely a scenario where they draw 50 tickets and none of them is yours.
Your probability of winning any one of those 50 tags is really (100% - (90% chance of losing ^ 10)) = 0.65, or 65%.
Note the example above is for independent draws meaning they put the winning ticket back in after each draw. But the logic also works for non-independent draws where the ticket is not put back in (ie. you can’t win multiple times) - you just need to adjust the probability of NOT winning for each subsequent draw and multiply those together instead of raising the original probability of NOT winning the first tag to the power of the number of tags. That would be (100% - (10/100 * 10/99 * 10/98 * 10/97 * … * 10/50)).