Okay, enough of the
preachy environmentalist stuff. I have a real problem, sent to me from Cousin Eric:
We all know the game Battleship, but Sophie has taken it to another level in an unheard of episode of beginners luck.
The past few weeks Sophie has graduated from Dominoes and Candyland to
Trouble (with the pop-o-matic) and now Battleship. She's not fared too
well with Trouble, but is undefeated (5-0) in Battleship and on Sunday
she took the game to a new level, one that I have never seen before
(and I've been playing since the 1960's). It took Sophie only 18
turns get the 16 hits to sink my entire fleet! With 100 pegs on the
board and only 16 hits available to opposing players the odds of
getting one hit at the beginning of the game is 6.25 to 1 or the
obvious 16%. The odds change slightly as each player takes a turn, but
the odds of hitting all the other players ships in only 18 turns
is.....I have no idea, but it's pretty crazy!
Was there foul play involved? I can find no evidence. Sophie's Aunt
Laury was in the room at the time, but she was on Sophie's side of the
room and board. Also, if Sophie was getting my coordinates from a
Sophie sympathizer (Laury, Molly, Rose or even Otis), at not yet 5
years of age her poker face has not yet evolved to keep this intel to
herself. So, all I can conclude is this unprecedented once in a
lifetime Battleship rout is legit and thought this was news worthy
enough to share it with you.
The picture below was taken right after the final shot was fired.
Wow. 16 out of 18 hits. Is this possible? No, because it actually takes 17 hits to sink everything. But 17 out of 19 hits? Is THIS possible? Yes, obviously, it is possible. That's a dumb question. The real question is: what are the odds? Aha! Time for math! (PS: I really need to get back to work)
We calculate the odds by calculating, after each turn, the likelihood of getting another hit. If the previous hit sank a ship, or this is the first shot, we calculate the odds of hitting a ship as (remaining open pegs in a ship)/(total pegs remaining). In Eric's example above, at the beginning of the game, there are 17 possible spots to hit a ship, and 100 total pegs, so the odds of hitting a ship are 17%. If the carrier is sunk, these odds change to 12/95, or 12.6%.
If the previous hit did not sink a ship, then the odds will be calculated depending on the number of pegs already in the ship, the type of ship we are trying to sink, and the position of that ship on the board. We will work through the carrier as an example.
You have just hit a ship. It is the carrier, although you have no way of knowing this. You just see a peg sticking out of the board. Your next move is to choose one of the 4 surrounding peg locations. What are your odds of hitting again?
The carrier has five locations. If you have hit either end of the carrier to start, your odds are 1/4. If you hit somewhere in the center, your odds are 2/4, or 1/2. So, your chances of hitting again are (1/4 + 1/2 + 1/2 + 1/2 + 1/4) / 5 = 40%.
Now you have two locations hit. What are your odds of hitting a third time? If the two hits are at the ends, the chances are 1/2. If the two hits are in the center, your odds are actually 100%. (1/2 + 1 + 1 + 1/2) / 4 = 75%.
Fourth time? (1/2 + 1 + 1/2) / 3 = 67%.
Fifth time?? (1/2 + 1/2) / 2 = 50%.
Now, we can calculate the odds of wiping out the cruiser without a miss, once you get that first hit: (0.4 * 0.75 * 0.67 * 0.5) = 10.05%.
If the ship is butted up against an edge, these odds are modified.
So, how do we distill all these facts down into the odds of this specific outcome? With computers! First, we look at the picture again, and assign each ship a position type: center of the board, longways against edge, shortways against edge, or corner.
carrier: center of board
battleship: center of board
submarine: center of board
destroyer: longways against edge
patrol boat: corner
Next, we'll fire up ye olde computer to crank out the odds of successfully hitting everything without missing (we'll get to the issue of the two misses in a second).
The odds of hitting all the spots without a single miss? 1.7 billion to one against. Really.
Okay, so how do the two misses affect the outcome? Not much. If we modify the original calculations so that there are 98 spaces to begin instead of 100, the odds drop to 1.5 billion to one against.
Oh, Sophie, you had a good thing going with your 5-0 record, but clearly you got greedy. Buy that girl a deck of cards, Eric, because her poker face is better than you think.
