Assume a standard $52$ card deck (no jokers) well shuffled. I was wondering how many cards would need to be randomly drawn from it (without replacement) on average to get about a $50$% chance of getting all $13$ cards in exactly one suit. For example, getting all $13$ of the diamond cards. How do you solve for this type of problem?
For clarification, a "winning" draw would be when one of the four suits is completely filled (all $13$ ranks of that suit are in the drawn cards). Any of the $4$ suits can be filled then the # of cards in the draw would be # of interest. There will be no cases where more than $1$ suit is completely filled in the hand. So for example, if the first $13$ cards drawn are all hearts, then hand stops at $13$ cards. If $12$ cards of each suit are drawn ($48$ cards total), then the hand would stop at the $49$th card draw (thus completing one of the suits in full).