Calculating Probability of Non-Capturing Rooks on a Chessboard

In summary, the conversation discusses the probability of placing 8 rooks on a chessboard so that none of them can capture each other. The initial attempt at a solution considers the total number of squares on the board and the number of empty squares for placement. However, it is incorrect. The correct approach involves considering the placement of one rook at a time, taking into account the restrictions on capturing. The probability is approximately .911 x 10^-5. The conversation also touches on the difference between the first statement and the second statement in terms of capture possibility.
  • #1
fk378
367
0

Homework Statement


If 8 rooks (castles) are randomly places on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.


The Attempt at a Solution


I just started it by knowing there are 64 squares on a chessboard. If there are 8 rooks, then that leaves 56 empty blocks.

Where to go from here?
 
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  • #2
How many different ways to put the rooks on the board without the no-capture restriction? How many ways with the restriction?
 
  • #3
no-capture restriction: 8! ways
without the restriction: 56! ways
 
  • #4
Not to give it away, but so you can check: P(no capture) [tex]\approx .911\times 10^{-5}[/tex].
 
  • #5
I don't understand how that is? Can you explain?
 
  • #6
Can you start by explaining why you think 56! is the number of ways to place them without restriction? That is certainly wrong.
 
  • #7
First, place a rook in a random spot on the board. How many ways are there to do this? Next, figure out how many places are left on the board where the second rook can't take the first rook. Now, how many ways are there to place another rook on the board so that it can't take either of the first two. Continue this until you get to the last rook (there should only be one space left for that one). Finally, figure out how many possible ways there are to place the 8 rooks on the board with no restrictions.
 
  • #8
2w7mm89.jpg


So you need one rook in one row.

[tex]P(A)=\frac{m}{n}[/tex]

8th row - the rook can move on 8! ways

Can you find the probability now?

Notice: you got 8 rooks so n=8 * ?
 
  • #9
fk378 said:

Homework Statement


If 8 rooks (castles) are randomly places on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

The first sentence is not equivalent to the second. :tongue:
 
  • #10
epenguin said:
The first sentence is not equivalent to the second. :tongue:

Why not?
 
  • #11
Because capture possibility depends on the colour of the rooks. :smile:
 

1. What is the role of probability in chess?

In chess, probability helps players make strategic decisions by determining the likelihood of potential outcomes based on the current board position and possible moves. It also helps players anticipate their opponent's moves and make counter-moves accordingly.

2. How is probability calculated in chess?

Probability in chess is typically calculated using mathematical models, such as Markov chains, that take into account the current board state, possible moves, and potential outcomes. These models can be quite complex and require a lot of computation, but they can provide valuable insights for players.

3. Can probability be used to predict the outcome of a chess game?

While probability can give players an idea of the likelihood of certain outcomes, it cannot accurately predict the outcome of a chess game. This is because chess is a game of perfect information, meaning that both players have access to all the necessary information to make informed decisions. Therefore, the outcome of a game depends on the players' skill and strategy, not just probability.

4. Are there any specific chess openings or strategies that rely heavily on probability?

There are certain chess openings and strategies that are known to involve a higher level of risk and uncertainty, making them more reliant on probability. For example, the Sicilian Defense opening is known for its complexity and unpredictability, as it involves sacrificing pieces and taking calculated risks. Other aggressive openings, such as the King's Gambit, also rely on probability to create attacking opportunities.

5. How can understanding probability improve one's chess game?

By understanding probability, players can make more informed decisions and anticipate their opponent's moves. This can help players create more strategic and efficient game plans, as well as identify potential weaknesses in their opponent's position. Additionally, understanding probability can also help players make calculated risks and take advantage of favorable odds.

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