Algebra.Com's Answer #838809 by ikleyn(52788)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Topic : Discrete Random Variable (Geometric Distribution)
\n" );
document.write( "Repeated independent trials are carried out in which the probability of success in each trial is
\n" );
document.write( "0.66 . Correct to 3 significant figures, find the probability that the first success occurs:
\n" );
document.write( "(a) On the third trial
\n" );
document.write( "(b) On or before the second trial
\n" );
document.write( "(c) After the third trial
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "Let Y means successful trial; N means unsuccessful trial.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(a) In case (a), they want you find the probability of this event: NNY.\r\n" );
document.write( "\r\n" );
document.write( " So, 1st trial is N (with the probability 1-0.66 = 0.34);\r\n" );
document.write( " 2nd trial is N (with the probability 1-0.66 = 0.34);\r\n" );
document.write( " 3rd trial is Y (with the probability 0.66.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " The probability of the event NNY is P(NNY) = 0.34*0.34*0.66 = 0.076296 (precise value). ANSWER to question (a)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b) The favorable events are  Y  NY \r\n" );
document.write( " (two favorable events, and they, obviously, are disjoint).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " So, the probability under the problem's question is\r\n" );
document.write( "\r\n" );
document.write( " P = P(Y) + P(NY) = 0.66 + 0.34*0.66 = 0.8844 (precise value). ANSWER to question (b)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(c) After solutions (a) and (b), you are just prepared ENOUGH to understand that\r\n" );
document.write( "\r\n" );
document.write( " P = P(N) + P(NN) + P(NNN) + P(NNNY) + P(NNNNY) + P(NNNNNY) + . . . (infinite series).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " Next, first three terms of this infinite sum are\r\n" );
document.write( "\r\n" );
document.write( " P(N) = 0.34;\r\n" );
document.write( "\r\n" );
document.write( " P(NN) = 0.34*0.34 = 0.1156;\r\n" );
document.write( "\r\n" );
document.write( " P(NNN) = 0.34^3 = 0.039304.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " The following terms P(NNNY) + P(NNNNY) + P(NNNNNY) + . . . (infinite series) \r\n" );
document.write( " represent the sum of an INFINITE geometric progression with the first term a = 0.34^3*0.66\r\n" );
document.write( " and the common ratio of r = 0.34.\r\n" );
document.write( "\r\n" );
document.write( " So, the sum of these following terms is \r\n" );
document.write( "\r\n" );
document.write( " P(NNNY) + P(NNNNY) + P(NNNNNY) + . . . (infinite series) =  =  =  =  = 0.039304 (rounded).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " Finally, the answer to question (c) is this sum\r\n" );
document.write( "\r\n" );
document.write( " P = 0.34 + 0.1156 + 0.039304 + 0.039304 = 0.534208 (precise value), \r\n" );
document.write( "\r\n" );
document.write( " or 0.534, rounded as requested. ANSWER to question (c)\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |