document.write( "Question 1163342: A coin with a 2-cm diameter is dropped onto a sheet of paper ruled by parallel lines that are 3 cm apart. Which is more likely, that the coin will land on a line, or that it will not? \n" ); document.write( "
Algebra.Com's Answer #787418 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "A coin with a 2-cm diameter is dropped onto a sheet of paper ruled by parallel lines that are 3 cm apart.
\n" ); document.write( "Which is more likely, that the coin will land on a line, or that it will not?
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document.write( "Notice that the radius of the coin is 1 centimeter.\r\n" );
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document.write( "We should track the position of the center of the coin at the coordinate plane relative to the lines of the grid.\r\n" );
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document.write( "Actually, we can concentrate our attention tracking the position of the center inside one fixed cell of the grid.\r\n" );
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document.write( "    If the center of the coin is closer than 1 cm to some integer line of the grid, then the coin INEVITABLY will cover that line.\r\n" );
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document.write( "So, in the fixed grid cell, we should draw the square of the side size of 1 centimeter, centered at the center of this cell.\r\n" );
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document.write( "This square has the area of 1 square centimeter, while the basic square of the grid has the area of  3*3 = 9 square centimeters..\r\n" );
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document.write( "The small square is the \"favorable\" area: if the coin's center gets there, the coin will not cover any line of the grid.\r\n" );
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document.write( "And in opposite, if the coin's center is in the base square outside the small square, the coin will INEVITABLY cover at least one of the grid lines..\r\n" );
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document.write( "It means that the probability that the coin will not land on the line is \"%289-1%29%2F9\", and the probability that the coin will land on some line of the grid is  \"1%2F9\".\r\n" );
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document.write( "ANSWER.  The probability that the coin will land on some grid line is  \"8%2F9\",  and  \r\n" );
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document.write( "         the probability that the coin will not land on some grid line is  \"1%2F9\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is one of classic problems of the Elementary Geometric Probability theory.\r
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\n" ); document.write( "\n" ); document.write( "For other similar solved problem,  see the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Classic problem of Elementary Geometric Probability theory \r
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\n" ); document.write( "\n" ); document.write( "In my solution I assumed that the sheet of paper is classic graph paper, ruled by parallel and perpendicular lines
\n" ); document.write( "that are 3 cm apart making square grid.\r
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\n" ); document.write( "\n" ); document.write( "If there are only parallel lines and there are no perpendicular lines, then the answer is different,
\n" ); document.write( "but the logic of the solution remains the same.\r
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\n" ); document.write( "\n" ); document.write( "You consider then parallel lines and infinite stripes of the unit width, instead of squares.\r
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\n" ); document.write( "\n" ); document.write( "In this case,  the probability that the coin will land on some line is   \"2%2F3\",
\n" ); document.write( "while the probability that it will not land is the complement   \"1%2F3\".\r
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