document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=121858>Question 121858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A certain airplane has two independent alternators to provide electrical power. The probability that a given alternator will fail on a 1-hour flight is .02. What is the probability that \r<BR>\n" );
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document.write( "a) both will fail? <BR>\n" );
document.write( "b) Neither will fail?<BR>\n" );
document.write( "c) One or the other will fail? Show all steps carefully.\r<BR>\n" );
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document.write( "Could you kindly help me. This is new to me.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #89546 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=89546\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The probability of two independent events occurring is equal to the product of each of<BR>\n" );
document.write( "the independent events. And the probabilities of some outcome must total to be 1.00.<BR>\n" );
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document.write( "In the discussion below, P(event) means the probability of the event being discussed. <BR>\n" );
document.write( "For example, P(Alt #1 Fails) means the probability that Alternator #1 fails.<BR>\n" );
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document.write( "Let's apply the above rules to solving this problem.  Since the probability of a single Alternator<BR>\n" );
document.write( "failing is 0.02, then the probability of it not failing must be 0.98 because those are the<BR>\n" );
document.write( "only two outcomes possible and the total of all the outcomes for an Alternator must be 1.0<BR>\n" );
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document.write( "The same is true for each of the two Alternators. The probability of failure is 0.02 and the<BR>\n" );
document.write( "probability of working OK is 0.98 for each.<BR>\n" );
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document.write( "What are the possible outcomes on a one-hour flight? The four possibilities are:<BR>\n" );
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document.write( "(1) Alternator #1 Fails and Alternator #2 Fails<BR>\n" );
document.write( "(2) Alternator #1 Fails and Alternator #2 Works OK<BR>\n" );
document.write( "(3) Alternator #1 Works OK and Alternator #2 Fails<BR>\n" );
document.write( "(4) Alternator #1 Works OK and Alternator #2 Works OK<BR>\n" );
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document.write( "The probability of (1) occurring is just the product of the two probabilities. So the<BR>\n" );
document.write( "probability of both Alternators failing is:<BR>\n" );
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document.write( "P(both fail) = P(Alt #1 Fails) * P(Alt #2 Fails) = 0.02 * 0.02 = 0.0004<BR>\n" );
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document.write( "Next, let's look at the probability of (4) occurring. It also is the product of the two<BR>\n" );
document.write( "probabilities. So the probability of both Alternators working OK is:<BR>\n" );
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document.write( "P(both OK) = P(Alt #1 Works OK) * P(Alt #2 Works OK) = 0.98 * 0.98 = 0.9604<BR>\n" );
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document.write( "Next, let's look at the probability of (2) occurring. It also is the product of the two<BR>\n" );
document.write( "probabilities. So the probability of Alternator #1 failing and Alternator #2 working OK is:<BR>\n" );
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document.write( "P(Alt #1 Fail and Alt #2 OK) = P(Alt #1 Fails) * P(Alt #2 Works OK) = 0.02 * 0.98 = 0.0196<BR>\n" );
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document.write( "Finally, let's look at the probability of (3) occurring. It also is the product of the two<BR>\n" );
document.write( "probabilities. So the probability of Alternator #1 working OK and Alternator #2 failing is:<BR>\n" );
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document.write( "P(Alt #1 OK and Alt #2 Fails) = P(Alt #1 Works OK) * P(Alt #2 Fails) = 0.98 * 0.02 = 0.0196<BR>\n" );
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document.write( "In either case (2) or case (3) one of the Alternators fails and the other one works OK.<BR>\n" );
document.write( "The probability of that happening is the sum of the two probabilities for these two cases. So<BR>\n" );
document.write( "the probability for one Alternator failing is:<BR>\n" );
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document.write( "P(one fail, one OK) = P(Alt #1 Fail and Alt #2 Works OK) + P(Alt #1 Works OK and Alt #2 Fails)<BR>\n" );
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document.write( "Substituting the probabilities results in:<BR>\n" );
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document.write( "P(one fail, one OK) = 0.0196 + 0.0196 = 0.0392<BR>\n" );
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document.write( "In summary, the three answers you are looking for are:<BR>\n" );
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document.write( "(a) probability both will fail = 0.0004<BR>\n" );
document.write( "(b) probability neither will fail (both will work OK) = 0.9604<BR>\n" );
document.write( "(c) probability one will fail = 0.0392<BR>\n" );
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document.write( "The probability of some outcome (all four of the combinations we said are the only things<BR>\n" );
document.write( "that can happen) must total to be 1.0 because something must happen. So we can check our<BR>\n" );
document.write( "answers by adding them up and see if they total 1.0. By adding 0.0004, 0.9604, and 0.0392<BR>\n" );
document.write( "you do get an answer of 1.0 ... so we are correct. <BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better.<BR>\n" );
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