Probability of not finding a joker among 54 cards until the 54th draw? I manage a "Joker Poker Raffle" for a veterans club in Shelton, WA.  We start each round with 54 cards (4 suits @ 13 cards plus 1 mini Joker and 1 Jackpot Joker) sealed in identical envelopes.  Each Saturday we sell tickets and then draw one ticket from a tumbler.  The person with the winning ticket selects one envelope.  If the selected envelope contains the Joker, the Jackpot is awarded, if not the game goes on.  The current round has gone for 53 weeks without a Jackpot win.  Folk are asking me what are the odds.
I wonder if the question can be viewed as flipping a fair coin 53 times and getting 53 heads and then getting 1 tail? 
In addition, several players take the position that inasmuch as everybody knows that the last envelope contains the Joker, we should end this round and start a new round with 54 new envelopes.  Their thinking is that the first 53 draws posed dual risk; i.e. the risk that one's ticket would be selected plus the risk that the winning ticket holder would select the envelope with the Joker.  Obviously, on the 54th draw, there is only singular risk.
Can you mathematicians help us veterans?  (A)  Is not identifying the Joker until the 54th draw a rare event, or can we expect this to happen frequently and (B) Is it fair to allow the person whose ticket is selected next Saturday to claim the Jackpot, or should we start a new round with 54 envelopes?
Thank you
Brian Walsh    
 A: The probability of not obtaining the joker until the 54th draw is $$\frac{53}{54}\times\frac{52}{53}\times\frac{51}{52}\times\cdots\times\frac{2}{3}\times\frac{1}{2}=\frac{1}{54}\approx0.02$$ which I would think is quite rare. 
As for the second question, sure, the wait was long enough...
A: 
(A) Is not identifying the Joker until the 54th draw a rare event, or can we expect this to happen frequently and (B) Is it fair to allow the person whose ticket is selected next Saturday to claim the Jackpot, or should we start a new round with 54 envelopes?

(A) The Joker can appear in any of the $54$ places, with equal probability, so there is simply a $1$ in $54$ chance that it appears last. It's fun that it happened to be the $54$th card, but it isn't particularly unusual—after all, some card has to be last!
As an example of something rare, suppose that the envelopes happened to have the cards in order ($\text{A}\spadesuit$ in the first envelope, then $2\spadesuit$, then $3\spadesuit$, etc.). The chance of that happening is $1$ in $230843697339241380472092742683027581083278564571807941132288000000000000$, which is exceedingly, almost unimaginably, rare.
(B) It is entirely fair! The prize needs to won at some point, and you've reached that point.
