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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

Brian Walsh
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    Once the evelopes are sealed, they remain sealed until the jackpot is awarded, correct? (i.e. you do not open all the envelopes, and re-stuff between draws). Then the Jacpot Joker is as likely as any other card to be in the $54^{th}$ envelope. (or $\frac 1{54}$) – Doug M Sep 26 '17 at 18:31
  • What is the meaning of the mini-joker ? – Peter Sep 26 '17 at 18:32
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    Do you keep the 54 envelopes undisturbed and with everyone knowing which envelopes were opened each week as the game proceeds? – Srini Sep 26 '17 at 18:32
  • "Fair" is subjective. If I get the winning ticket on the first week, I have a $1/54$ chance of getting the jackpot. If I get the winning ticket when there are only two envelopes left, I have a $1/2$ chance of getting the jackpot. You could argue that it's unfair to the people who bought the tickets the first week, since pay the same for a lower chance, so every week already gives you better chances. You could argue that buying tickets on earlier weeks in fair anyway, since those people get a shot at it before the people who wait a few weeks before buying tickets. – Kevin Long Sep 26 '17 at 18:37
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    @KevinLong The person who will win the jackpot will consider the game to be fair no matter how this happened So yes, "fair" is in fact subjective! :) – Peter Sep 26 '17 at 18:39
  • And someone never winning in such games might consider it to be "fair" if such lotteries do not at all exist because then, the chances to win, are equal for everyone (namely $0$) :) – Peter Sep 26 '17 at 18:42

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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...

John Doe
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