![\"\"](\"/images/stars/stars-13.gif\")
You can put this solution on YOUR website!
\r\n" );
document.write( "When you see the words \"at least one\", we know to first find the probability of\r\n" );
document.write( "the complement event and then subtract from 1.\r\n" );
document.write( "\r\n" );
document.write( "The complement event is that no letters go in the right envelope.\r\n" );
document.write( "\r\n" );
document.write( "The denominator of the probability is obviously 5!.\r\n" );
document.write( "\r\n" );
document.write( "The numerator of the probability is what is difficult.\r\n" );
document.write( "\r\n" );
document.write( "Let an be the number of ways n letters can all put in n wrong envelopes.\r\n" );
document.write( "\r\n" );
document.write( "We want to find a5 \r\n" );
document.write( "\r\n" );
document.write( "Suppose the 5 letters are numbered 1,2,3,4,5, and the envelopes are also\r\n" );
document.write( "numbered 1,2,3,4,5. We have to find the number of ways in which no letter goes\r\n" );
document.write( "in the envelope having the same number as its number. There are 4 ways for the\r\n" );
document.write( "first letter to go in an envelope other than the 1st one. Let us assume, for\r\n" );
document.write( "definiteness, that the letter 1 gets put in envelope 3. There are now two\r\n" );
document.write( "possibilities:\r\n" );
document.write( "\r\n" );
document.write( "case 1. letter 3 goes in envelope 1,\r\n" );
document.write( "\r\n" );
document.write( "That leaves letters 2,4,5 to be put wrongly in envelopes 2,4,5. That's\r\n" );
document.write( "a3 possibilities.\r\n" );
document.write( "\r\n" );
document.write( "case 2. letter 3 does not go in envelope 1.\r\n" );
document.write( "\r\n" );
document.write( "That leaves letters 2,3,4,5 to put wrongly in envelopes 1,2,4,5. But since\r\n" );
document.write( "letter 3 cannot go in envelope 1, that will be the same as if we re-labeled\r\n" );
document.write( "letter 3 as letter X and re-labeled envelope 1 as envelope X. Then we would\r\n" );
document.write( "have letters 2,X,4,5 to go wrongly in envelopes X,2,4,5. That's a4\r\n" );
document.write( "possibilities.\r\n" );
document.write( " \r\n" );
document.write( "So the number of ways letter 1 could have gone in envelope 3 is \r\n" );
document.write( "a3 + a4 \r\n" );
document.write( "\r\n" );
document.write( "But we could have put letter 1 in any of the 4 letters {2,3,4,5}. Therefore we\r\n" );
document.write( "must multiply that by 4, and so we get\r\n" );
document.write( "\r\n" );
document.write( "a5 = 4(a3 + a4)\r\n" );
document.write( "\r\n" );
document.write( "By the same token\r\n" );
document.write( "\r\n" );
document.write( "a4 = 3(a2 + a3) \r\n" );
document.write( "\r\n" );
document.write( "It's obvious that a2 = 1 because if there are 2 letters and 2\r\n" );
document.write( "envelopes, then there is exactly 1 way to put letter 1 in envelope 2 and letter\r\n" );
document.write( "2 in envelope 1.\r\n" );
document.write( "\r\n" );
document.write( "It's also easy to discover that a3 = 2 because envelopes 1,2,3 could\r\n" );
document.write( "only either go respectively in envelopes 2,3,1 or respectively in envelopes\r\n" );
document.write( "3,1,2.\r\n" );
document.write( "\r\n" );
document.write( "Therefore, \r\n" );
document.write( "\r\n" );
document.write( "a4 = 3(a2 + a3) = 3(1+2) = 3(3) = 9\r\n" );
document.write( "\r\n" );
document.write( "and\r\n" );
document.write( "\r\n" );
document.write( "a5 = 4(a3 + a4) = 4(2+9) = 4(11) = 44\r\n" );
document.write( "\r\n" );
document.write( "So the probability that all the letters will go wrong is 44 ways out of all 5!\r\n" );
document.write( "ways they could have been placed. That probability is 
= 
= 
.\r\n" );
document.write( "\r\n" );
document.write( "Therefore the probability that at least one letter goes in the correct envelope\r\n" );
document.write( "is 1 - = 
\r\n" );
document.write( "\r\n" );
document.write( "Edwin \n" );
document.write( "