166 CHAPTER 4 Probability Rationale for the Multiplication Rule To see the reasoning that underlies the multiplication rule, consider a pop quiz consisting of these two questions: 1. True or false: A pound of feathers is heavier than a pound of gold. 2. Who said, “By a small sample, we may judge of the whole piece”? (a) Judge Judy; (b) Judge Dredd; (c) Miguel de Cervantes; (d) George Gallup; (e) Gandhi The answers are T (true) and c. (The first answer is true, because weights of feathers are in avoirdupois units where a pound is 453.59 g, but weights of gold and other precious metals are in troy units where a pound is 373.24 g. The second answer is from Don Quixote by Cervantes.) Here is the sample space for the different possible answers: Ta Tb Tc Td Te Fa Fb Fc Fd Fe If both answers are random guesses, then the above 10 possible outcomes are equally likely, so P1both correct2 = P1T and c2 = 1 10 = 0.1 With P1T and c2 = 1>10, P1T2 = 1>2, and P1c2 = 1>5, we see that 1 10 = 1 2 # 1 5 Go Figure 13: Current record for the number of folds of a sheet of paper; very thin paper was used and the sheet was almost 2 miles long. If you could fold a sheet of paper 42 times, it would have a height that would reach to the moon. If you could fold a sheet of paper 103 times, it would have a height roughly equal to the diameter of the observable universe. SOLUTION a. The failure rate of 2.89% converts to a probability of 0.0289 for failure, so the probability that the hard drive will not fail during the year is found as follows: P1hard drive does not fail2 = 1 - P1failure2 = 1 - 0.0289 = 0.971 b. With two independent hard drives, the computer will work if both hard drives don’t fail. That is, the computer will work if at least one of the hard drive works. P1at least one hard drive does not fail2 = 1 - P1both hard drives fail2 = 1 - 10.0289210.02892 = 0.999 INTERPRETATION With only one hard drive, there is a 0.0289 probability of failure, but with two independent hard drives, there is only a 0.001 probability that the computer will not work with at least one operating hard drive. By using two hard drives instead of just one, the risk of failure decreased by a factor of 1>28.9. By using two independent hard drives, the risk of a serious failure is dramatically decreased and reliability of the computer is dramatically increased. Redundancy at work! YOUR TURN. Do Exercise 25 “Redundancy in Computer Hard Drives.” Seagate Hard Drive: Redundancy for Better Reliability EXAMPLE 8 The Seagate hard drive model ST4000DM000 has a failure rate of 2.89% in a year (based on data from Backblaze, Inc.). a. If your computer has only the Seagate hard drive model ST4000DM000, what is the probability that it will work for a year? b. If your computer has two Seagate model ST4000DM000 hard drives, what is the probability at least one of them will work for a year?
RkJQdWJsaXNoZXIy NjM5ODQ=