Critical Line Volume 14
Dan Mayoh brings you the 14th instalment of the popular Critical Line Puzzle Challenge with a nod to a Royal Flush Jackpot.
A certain Melbourne poker room offers a progressive ‘Royal Flush Jackpot’ which goes up over time, and is awarded if your hand makes a royal flush on the flop when playing Texas Hold’em. This means that two cards in your hand plus the first 3 community cards are a 10, Jack, Queen, King and Ace of the same suit (any ordering of the 5 cards is permissible). Once the jackpot is awarded, it resets to a smaller amount and starts building again.
After not being awarded for so long that the jackpot reached its highest ever level, it was then awarded four times in less than ten days earlier this month. It got me thinking about the expectations involved, and forms the basis of this volume’s puzzle.
Assume that there is a poker room offering this jackpot to players and the following information holds true:
 There are 18 tables, each with 10 players, and each table deals 20 hands per hour. For this purpose, a ‘hand’ consists of all 10 players receiving 2 cards each, and 3 community cards being dealt in the centre of the table, all from a standard 52card deck that is shuffled after each hand.
 All tables operate nonstop 24 hours a day and are always full with exactly 10 players.
 The 18 * 20 = 360 hands dealt each hour are spread out in such a way that every 10 seconds there is a new hand dealt somewhere in the poker room.
 If any of the 10 players in a hand can make a royal flush using their two individual cards plus the 3 community cards, the jackpot is won.
 The jackpot increases at a rate of 15 cents every 10 seconds until it is won. When won, it resets to a value equal to 1/3 of the won amount.
 At midnight on the morning of 1 July 2017 the jackpot amount is exactly $10,000.
The challenge for this month is to figure out:
 What is the probability of the jackpot being won in a given hand?
 What is the expected length of time between jackpot wins?
 What is the expected value of the jackpot size at midnight on the night of 31 December 2017? Explain the reasoning behind your answer as elegantly as you can.
For your chance to win $50, send your solutions to ActuariesMag@actuaries.asn.au
Solution (Volume 13)
Lost in the jungle
An actuary has gone for a hike in the jungle and become disoriented. He knows that he is 1km away from a perfectly straight road of infinite length, but does not know which direction he is facing. The forest is very dense so he will not be able to see the road until he walks upon it. What is the shortest distance he needs to walk to be certain of finding the road?
Solution
We had seven submissions this month. The table below shows four possible routes to escape the jungle. The simplest strategy is to walk $$1$$ km in any direction and then walk $$2\pi$$ km in a circle around your starting position. This guarantees that you hit the road eventually.
The best strategy, submitted by Matthew Floyd, Phichol Lee and Andrew Parker, is to walk $$\frac{2}{\sqrt{3}}$$ in any direction, then to turn 150 degrees and walk $$\frac{1}{\sqrt{3}}$$ until you meet the circle of radius $$1$$km that is centred at your starting point. You follow the circle around for $$\frac{7}{6}\pi$$ km at which point you head straight for one $$1$$ km. This path will cross every tangent to the circle and therefore guarantee that you find the road.
Note: the red line in the illustrations below represents a tangent to the circle of radius $$1$$km that is centred at your starting position.
The winner of this month’s book voucher is Matthew Floyd.
Submission 
Strategy (Illustration) 
Distance (exact) 
Distance (approx.) 
John De Ravin, Chris Barry 
$$1 + 2\pi$$ 
$$7.283$$km 

Michael Eabry, Paul Swinhoe 
$$2 + \sqrt{2} + \pi$$ 
$$6.556$$km 
Nicola Westoby 
n/a 
$$6.459$$km 

Matthew Floyd, Andrew Parker, Phichol Lee

$$1 + \sqrt{3} + \frac{7}6 \pi$$ 
$$6.397$$km 
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