You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
system
In the game of Connect Four, there are seven vertical columns which have spaces for six tokens, forming a \(7 \times 6\) grid of spaces. Two players, White and Black, move alternately. A player takes a turn by picking a column that is not already full and dropping a token of their color into the lowest unoccupied space in that column. The game ends when one player has four consecutive tokens of their color in a line: either horizontally, vertically, or diagonally. The player who achieves this wins the game.

Assume that the two players play this game randomly. Each player, on their turn, picks a random column that is not full and drops a token of their color into that column. This process continues until one player wins or all columns are filled. Let \( P \) be the probability that all columns are filled without any player obtaining four tokens in a row of their color. Estimate \( P \). 

An estimate of \( E > 0 \) earns \( \lfloor 20 \min (P / E, E / P)\rfloor \) points.
user
0.0028
rule
In the game of Connect Four, there are seven vertical columns which have spaces for six tokens, forming a \(7 \times 6\) grid of spaces. Two players, White and Black, move alternately. A player takes a turn by picking a column that is not already full and dropping a token of their color into the lowest unoccupied space in that column. The game ends when one player has four consecutive tokens of their color in a line: either horizontally, vertically, or diagonally. The player who achieves this wins the game.

Assume that the two players play this game randomly. Each player, on their turn, picks a random column that is not full and drops a token of their color into that column. This process continues until one player wins or all columns are filled. Let \( P \) be the probability that all columns are filled without any player obtaining four tokens in a row of their color. Estimate \( P \). 

An estimate of \( E > 0 \) earns \( \lfloor 20 \min (P / E, E / P)\rfloor \) points.
train