document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169076>Question 1169076</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You and your friend are taking the driving test to obtain your driver’s licenses. The<BR>\n" );
document.write( "probability for you to pass the test is 0.9 and for your friend it is 0.8. Calculate the<BR>\n" );
document.write( "probability that you two make a total of 8 attempts to obtain your licenses.<BR>\n" );
document.write( "(if you got your license on the 2nd attempt and your friend got his on 3rd attempt, you<BR>\n" );
document.write( "made a total of 5 attempts) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851450 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851450\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $X$ be the number of attempts you take to pass the driving test, and $Y$ be the number of attempts your friend takes to pass the driving test. We are given that the probability of you passing on any attempt is $p_1 = 0.9$, and the probability of your friend passing on any attempt is $p_2 = 0.8$. The attempts are independent for both individuals.\r<BR>\n" );
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document.write( "The number of attempts for each person follows a geometric distribution. The probability mass function for a geometric distribution is given by $P(K=k) = (1-p)^{k-1}p$, where $k$ is the number of trials until the first success, and $p$ is the probability of success on each trial.\r<BR>\n" );
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document.write( "In this case, for you:<BR>\n" );
document.write( "$P(X=i) = (1-0.9)^{i-1}(0.9) = (0.1)^{i-1}(0.9)$, for $i = 1, 2, 3, \dots$\r<BR>\n" );
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document.write( "For your friend:<BR>\n" );
document.write( "$P(Y=j) = (1-0.8)^{j-1}(0.8) = (0.2)^{j-1}(0.8)$, for $j = 1, 2, 3, \dots$\r<BR>\n" );
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document.write( "We want to find the probability that the total number of attempts is 8, i.e., $P(X+Y=8)$. Since the attempts are independent, we can write this as the sum of the probabilities of all pairs $(i, j)$ such that $i+j=8$:\r<BR>\n" );
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document.write( "$P(X+Y=8) = \sum_{i=1}^{7} P(X=i \text{ and } Y=8-i) = \sum_{i=1}^{7} P(X=i) P(Y=8-i)$\r<BR>\n" );
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document.write( "Now, we substitute the probability mass functions:<BR>\n" );
document.write( "$P(X+Y=8) = \sum_{i=1}^{7} [(0.1)^{i-1}(0.9)] [(0.2)^{(8-i)-1}(0.8)]$<BR>\n" );
document.write( "$P(X+Y=8) = \sum_{i=1}^{7} (0.9)(0.8) (0.1)^{i-1} (0.2)^{7-i}$<BR>\n" );
document.write( "$P(X+Y=8) = 0.72 \sum_{i=1}^{7} (0.1)^{i-1} (0.2)^{7-i}$\r<BR>\n" );
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document.write( "Let $k = i-1$, so as $i$ goes from 1 to 7, $k$ goes from 0 to 6. The sum becomes:<BR>\n" );
document.write( "$\sum_{k=0}^{6} (0.1)^{k} (0.2)^{7-k}$<BR>\n" );
document.write( "$= (0.1)^0 (0.2)^7 + (0.1)^1 (0.2)^6 + (0.1)^2 (0.2)^5 + (0.1)^3 (0.2)^4 + (0.1)^4 (0.2)^3 + (0.1)^5 (0.2)^2 + (0.1)^6 (0.2)^1$<BR>\n" );
document.write( "$= 1 \times 0.0000128 + 0.1 \times 0.000064 + 0.01 \times 0.00032 + 0.001 \times 0.0016 + 0.0001 \times 0.008 + 0.00001 \times 0.04 + 0.000001 \times 0.2$<BR>\n" );
document.write( "$= 0.0000128 + 0.0000064 + 0.0000032 + 0.0000016 + 0.0000008 + 0.0000004 + 0.0000002$<BR>\n" );
document.write( "$= 0.0000254$\r<BR>\n" );
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document.write( "Now, multiply by 0.72:<BR>\n" );
document.write( "$P(X+Y=8) = 0.72 \times 0.0000254 = 0.000018288$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{0.000018288}$ </SPAN>\n" );
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