document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164737>Question 1164737</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A trading company has eight computers that it uses to trade on the New York Stock Exchange<BR>\n" );
document.write( "(NYSE). The probability of a computer failing in a day is 0.005, and the computers fail<BR>\n" );
document.write( "independently. Computers are repaired in the evening and each day is an independent trial.<BR>\n" );
document.write( "(a) What is the probability that all eight computers fail in a day?<BR>\n" );
document.write( "(b) What is the mean number of days until a specific computer fails?<BR>\n" );
document.write( "(c) What is the mean number of days until all eight computers fail in the same day\r<BR>\n" );
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document.write( "compute the solution in R ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853522 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=853522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this, we will use the properties of the **Binomial Distribution** (for successes in a fixed number of trials) and the **Geometric Distribution** (for the time until an event occurs).\r<BR>\n" );
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document.write( "### Mathematical Breakdown\r<BR>\n" );
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document.write( "**Part (a): Probability that all eight computers fail in a day**<BR>\n" );
document.write( "Since each computer is an independent trial with a constant probability of failure , the number of failures  follows a Binomial distribution: .<BR>\n" );
document.write( "The probability of all 8 failing is:\r<BR>\n" );
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document.write( "**Part (b): Mean number of days until a specific computer fails**<BR>\n" );
document.write( "The number of days until a specific event occurs follows a Geometric distribution. The mean (expected value) is given by:\r<BR>\n" );
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document.write( "**Part (c): Mean number of days until all eight computers fail on the same day**<BR>\n" );
document.write( "In this case, we treat the \"event\" as the simultaneous failure of all 8 computers. Let the probability found in part (a) be . The mean number of days until this specific event occurs is:\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### R Solution\r<BR>\n" );
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document.write( "You can copy and paste the following code into your R console or RStudio:\r<BR>\n" );
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document.write( "```r<BR>\n" );
document.write( "# Define parameters<BR>\n" );
document.write( "n <- 8<BR>\n" );
document.write( "p <- 0.005\r<BR>\n" );
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document.write( "# (a) Probability that all eight computers fail in a day<BR>\n" );
document.write( "# Using the dbinom function or direct calculation<BR>\n" );
document.write( "prob_all_fail <- dbinom(8, size = n, prob = p)<BR>\n" );
document.write( "# prob_all_fail <- p^8\r<BR>\n" );
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document.write( "cat(\"(a) Probability all 8 fail in a day:\", format(prob_all_fail, scientific = TRUE), \"\n\")\r<BR>\n" );
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document.write( "# (b) Mean number of days until a specific computer fails<BR>\n" );
document.write( "# Mean of Geometric distribution is 1/p<BR>\n" );
document.write( "mean_days_specific <- 1 / p\r<BR>\n" );
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document.write( "cat(\"(b) Mean days until a specific computer fails:\", mean_days_specific, \"days\n\")\r<BR>\n" );
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document.write( "# (c) Mean number of days until all eight computers fail in the same day<BR>\n" );
document.write( "# Mean is 1 over the probability of the event occurring<BR>\n" );
document.write( "mean_days_all_fail <- 1 / prob_all_fail\r<BR>\n" );
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document.write( "cat(\"(c) Mean days until all 8 fail on same day:\", format(mean_days_all_fail, big.mark = \",\"), \"days\n\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "### Explanation of Results\r<BR>\n" );
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document.write( "* **(a)** The probability is . This is effectively zero, making it an extremely rare \"black swan\" event.<BR>\n" );
document.write( "* **(b)** On average, you can expect a specific computer to fail once every **200 days**.<BR>\n" );
document.write( "* **(c)** The mean wait time for all 8 to fail at once is ** days**. For perspective, the universe is approximately  days old, so this event is statistically unlikely to ever happen in human history.\r<BR>\n" );
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document.write( "Would you like to see how the probability changes if the computers were not independent (e.g., a power surge affecting all of them at once)? </SPAN>\n" );
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