Home > Probability
|Treasure Island (Posted on 2010-08-15)
It is the Year of Our Lord 1766. After many adventures, the good guys and the pirates have found a cave containing 3 brassbound chests. Each chest contains 1000 coins, which may be gold doubloons, silver pieces of eight (poe) or base metal (duds). Doubloons are worth 100 each, poe 10, and duds 1.
Flint’s chest contains 400 doubloons, 500 poe, and 100 duds (well, nobody’s perfect).
Tuff’s chest contains 100 doubloons, 400 poe and 500 duds.
Stonibroke’s chest contains 0 doubloons, 100 poe and 900 duds.
Apart from this, the chests are identical, and no-one knows which is which at the outset.
After lengthy ‘discussion’ outside the cave at pistol-point, the following ‘fair’ means of dividing the treasure is arranged:
1. In turn the good guys, then the pirates, may nominate chest 1,2, or 3. ‘Honest’ Long John Silver will enter the cave, then draw a coin at random from that chest, show it to everyone, and return it to that chest.
2. Alternatively, in its turn, either side may choose a chest. The other side may then have as many additional draws as it likes before choosing one of the other two chests. Each side then gets the chest they chose, and the remaining chest is donated to the BGFSS (Ben Gunn Fund for Shipwrecked Seamen).
Seems straightforward enough, me hearties? There’s one snag.
‘Honest’ Long John shares as a pirate, and is also on the make for himself. Naturally, on first entering the cave, he will rummage through the chests to see which contains the most gold.
He has accordingly arranged this signal with his pirate shipmates:
- On any good guys’ turn, if he exhibits a doubloon, this means that the chest is Flint’s.
- Any doubloons drawn from Tuff’s chest on such turns will be palmed by Silver and form his personal reward; a dud will be exhibited instead.
- All other draws will be honestly conducted and reported.
The pirates will wait for Silver’s signal before choosing a chest.
Jim of the good guys, hiding in a coconut tree, has overheard the pirates' arrangement, thus confirming the suspicions of the good guys' leader, Dr. Livesey FRS, concerning Silver's duplicity.
What chance do the good guys have of getting their hands on Flint’s chest, if their first draw is a poe?
Bonus: What are their chances if they haven't drawn at all yet?
No Solution Yet
Submitted by broll
You must be logged in to post comments.)
There are no comments yet.
Please log in:
About This Site
New Comments (6)
Top Rated Problems
This month's top
Most Commented On