document.write( "Question 1205466: The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are chosen at random in the interval $[0,1].$ Let $I$ be the interval between $x_1$ and $x_2,$ and let $J$ be the interval between $x_3$ and $x_4.$ Find the probability that intervals $I$ and $J$ both have length greater than $3/4$. \n" ); document.write( "

Algebra.Com's Answer #842290 by ikleyn(52914) $\"\"$ $\"About$

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\n" ); document.write( "The numbers x_1, x_2, x_3, x_4 are chosen at random in the interval [0,1].
\n" ); document.write( "Let I be the interval between x_1 and x_2, and let J be the interval between x_3 and x_4.
\n" ); document.write( "Find the probability that intervals I and J both have length greater than 3/4.
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\n" ); document.write( "\n" ); document.write( "        The problem is posed/worded not perfectly (and far from to be perfect).
\n" ); document.write( "        I will not even try to re-formulate it in a perfectionist's way.
\n" ); document.write( "        I will simply assume that it is formulated perfectly by somebody else,
\n" ); document.write( "        and will solve it for this \"ideal\" case.\r
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document.write( "In order for x_1 and x_2 satisfy  the condition \r\n" );
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document.write( "    \"[x_1,x_2} is an interval and the length of the interval [x_1,x_2] is greater than 3/4\",\r\n" );
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document.write( "these conditions must be satisfied\r\n" );
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document.write( "    x_1 <= 1/4;   (1)\r\n" );
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document.write( "    x_1 <= x_2,  x_2-x_1 >= 3/4.    (2)\r\n" );
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document.write( "x_2 |\r\n" );
document.write( "    | \r\n" );
document.write( "8/8 + . . . . .|---|---|---|---|---|\r\n" );
document.write( "    | . . . \r\n" );
document.write( "7/8 + . .\r\n" );
document.write( "    | .\r\n" );
document.write( "6/8 +  \r\n" );
document.write( "    |\r\n" );
document.write( "5/8 +\r\n" );
document.write( "    |\r\n" );
document.write( "4/8 +\r\n" );
document.write( "    |\r\n" );
document.write( "3/8 +\r\n" );
document.write( "    |\r\n" );
document.write( "2/8 +\r\n" );
document.write( "    |\r\n" );
document.write( "1/8 +\r\n" );
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document.write( "    +---|---|---|---|---|---|---|---|\r\n" );
document.write( "  0    1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 <<<---x_1\r\n" );
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document.write( "In order for you could see the domain, which is good for points (x_1,x_2)) under conditions (1) and (2),\r\n" );
document.write( "I prepared this plot above. It has horizontal axis x_1 and vertical axis x_2.\r\n" );
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document.write( "The points that satisfy conditions (1) and (2), are the point close to the upper left corner of the square \r\n" );
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document.write( "    [0 <= x_1 <= 1, 0 <= x_2 <= 1].\r\n" );
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document.write( "It is the right-angled triangle with the legs 1/4 in each direction, so the area of this triangle is  = .\r\n" );
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document.write( "Thus the probability to get a \"good\" point (x_1,x_2) randomly is 1/32.\r\n" );
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document.write( "The probability to get the other \"good\" point (x_3,x_4) randomly is AGAIN 1/32.\r\n" );
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document.write( "Therefore, since the events to get a \"good\" point (x_1,x_2) and a \"good\" point (x_3,x_4) are independent,\r\n" );
document.write( "the probability under the problem's question is the product \r\n" );
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document.write( "    P =  =  = .    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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