Odds of never missing in a game of Battleship

17 hits wins at Battleship. My mate and I were arguing which would be harder, never missing a single hit at Battleship (going 17 for 17) or winning the lottery. What are the odds of never missing your opponent at Battleship?

I'll take a whack at this, see if any of this math stuff is sinking in....

I think if you worded the question as "randomly selecting numbers" and never missing, it could be figured as (17/100)*(16/99)*(15/98)...etc and so on (assuming there's 100 spots, I can't remember for sure). The problem I have is Battleship is a strategy game, where if you get one hit, instead of selecting a random # out of the 99 left, you'd typically select only one number out of the 1-4 remaining spots adjacent to that hit. When it comes to figuring THAT mess out, or how you'd incorporate the fact that it may be a 2 hit PT boat or a 5 hit Carrier, I'm beyond out of my league.

I'd be interested if ME, Crystal, Wiz, Miplet, DG, or any other of our math genius' had an answer for this one.

Assume that your strategy is to hit a ship and then attempt to sink the ship.

Your odds of hitting the first ship is indeed 17/100. At that point, you usually have 4 choices as to which direction to go. But if the hit is on an edge, then you could have only 3 or 2 directions to go. And at the same time, if your hit is on the middle of the ship then you are only in 2 directions.

My ball park guess is about 1 million to one.

Face has the right formula - for completely blind / random hits.
It works out to 0.000000000000000000150372890040097 or 1 in 6,650,134,872,937,200,000.
Note that this is NOT an exact number. Excel rounded it.

One you start to consider strategy and edges, man it gets complex.

Quote: boymimbo

Assume that your strategy is to hit a ship and then attempt to sink the ship.

Your odds of hitting the first ship is indeed 17/100. At that point, you usually have 4 choices as to which direction to go. But if the hit is on an edge, then you could have only 3 or 2 directions to go. And at the same time, if your hit is on the middle of the ship then you are only in 2 directions.

My ball park guess is about 1 million to one.

The way I remember Battleship being played, if you had 17 rounds available, you did not get to fire one, see whether it hit a ship or not, and then fire again. You had to fire all 17 rounds and were told how many, if any, were hits -- you weren't even told which shots were hits. Am I remembering that correctly? If so, then the likelihood of the initial 17 rounds (with no feedback yet) hitting the exact 17 spots that contained ship parts would be the extremely low figure already quoted. (Haven't calculated it myself.)

Ten months ago, I made this post about Battleship in a thread discussing whether video poker games are all dealt randomly. I noted that my experience with an early version of video Battleship led to my continuing suspicion (justified or not) about the trustworthiness of video craps that I had seen in a tribal casino. That suspicion seems quite relevant these days in light of the various BLR/on-line craps threads.

I don't have much.

I looked at placing the single largest boat (5 hits). It has 120 ways to place it randomly on the board. If the placement is completely random, then the best first move is to select one of the four most inner squares. If that hits, then randomly select up/down/left/right. If that hits, then select the opposite direction. If that hits, the last two selections are 50/50. If you continue to hit spots and always make the best choice, then the probability of sinking the large boat with the first 5 hits is 1 in 1680.

Interesting question, but one that you will likely not get an answer to.

Quote: Doc

The way I remember Battleship being played, if you had 17 rounds available, you did not get to fire one, see whether it hit a ship or not, and then fire again. You had to fire all 17 rounds and were told how many, if any, were hits -- you weren't even told which shots were hits. Am I remembering that correctly? If so, then the likelihood of the initial 17 rounds (with no feedback yet) hitting the exact 17 spots that contained ship parts would be the extremely low figure already quoted. (Haven't calculated it myself.)

That's a good point - the rules really determine everything. I haven't played Battleship in ages, but I'm sure I remember having red pegs and white pegs and knowing whether a shot was a hit or miss after making it. "You sunk my Battleship!"

Computing the odds of an intelligent player (the hypothetical perfect logician) under that scenario is difficult, not only because of the choices available to the player but because of the strategy that may be used by the opponent. For example, I used to T two of my ships together. In other words, you can't assume the ships are placed randomly because that may not be optimal if the opponent is attempting to maximize your number of shots (which they are). So then you're into game theory...

Or, you could be playing against an alien intelligence. I have no idea what the odds of winning are there...

The rules as I remember them were that you got feedback on every shot. Call a position, and you got "miss, hit, sunk." You were NOT told what type of ship you hit or sunk.

Quote: MathExtremist

That's a good point - the rules really determine everything. I haven't played Battleship in ages, but I'm sure I remember having red pegs and white pegs and knowing whether a shot was a hit or miss after making it. "You sunk my Battleship!"

It's a very simple game:

You call out a square by using its coordinates. If you hit, you place a red peg on your board and, if memory serves, you keep going until you miss; your opponent places a red peg on the ship hit. If you miss, you place a white peg on your board and your turn is over; the opponent palces a white peg where the shot missed. Each ship is sunk when all the squares it occupies are hit. The first player to sink all the opponent's ships wins.

The problem, as usual, is the honesty of the players. I did know people who'd cheat at Battleship, Mastermind and Go Fish.

There are two ways I remember playing it. According to the official battleship rules from my trusted game downstairs:

Quote: Hasbro

It's a hit!

If you call out a shot location that is occupied by a ship on your opponent's ocean grid, your shot is a hit! Your opponent tells you which ship you have which ship you have hit (destroyer, submarine, etc.). Record your hit by placing a red peg in the corresponding target hole on your target grid.".

You get one shot per turn.

I've played two variants:

(1) you are not told what ship you hit, and/or
(2) after a hit, you are allowed to continue selection until you get a miss.