Lesson Probability for a computer to be damaged by viruses
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<H2>Probability for a computer to be damaged by viruses</H2> <H3>Problem</H3>Three computer viruses arrived as an e-mail attachment. Virus A damages the system with probability 0.4. Independently of it, virus B damages the system with probability 0.5. Independently of A and B, virus C damages the system with probability 0.2. What is the probability that the system gets damaged? <B>Solution</B> <pre> Since virus A damages the system with probability 0.4, the computer is not damaged by the virus A with probability 1-0.4. Since virus B damages the system with probability 0.5, the computer is not damaged by the virus B with probability 1-0.5. Since virus C damages the system with probability 0.2, the computer is not damaged by the virus C with probability 1-0.2. Since viruses A, B, and C act independently, probability that the computer is not damaged by the viruses A, B and C is the product (1-0.4)*(1-0.5)*(1-0.2). Hence, probability that the computer is not damaged by the viruses A, B and C is the complement 1 - (1-0.4)*(1-0.5)*(1-0.2) = 1 - 0.6*0.5*0.8 = 1 - 0.24 = 0.76. <B>Answer</B>. The probability that the system gets damaged is 0.76. </pre> <H3>Problem 2</H3>Successful implementation of a new system is based on three independent modules. Module 1 works properly with probability 0.96. For modules 2 and 3, these probabilities equal 0.95 and 0.90. Compute the probability that at least one of these three modules fails to work properly. <B>Solution</B> <pre> Do it backwards, find the probability that none fail to work. P = 0.96*0.95*0.90, P = 0.8208. The complement of this calculated probability is that at least one of the modules fails to work. So, P = 1-0.8208, P = 0.1792. <B>Answer</B>. the probability that at least one of these three modules fails to work properly is 0.1792. </pre> <H3>Problem 3</H3>In a home theater system, the probability that the video components need repair within 1 year is 0.03, the probability that the electronic components need repair within 1 year is 0.007, and the probability that the audio components need repair within 1 year is 0.004. Assuming that the events are independent, find the following probabilities : (a) At least one of these components will need repair within 1 year; (b) Exactly one of these component will need repair within 1 year. <B>Solution</B> <pre> (a) At least one of these components will need repair within 1 year. According to the condition, the probability that V-component will not need a repair within 1 year is (1-0.03), E-component will not need a repair within 1 year is (1-0.007), A-component will not need a repair within 1 year is (1-0.004). Hence, the probability that NO ONE of the three components will not need a repair is the product (1-0.03)*(1-0.007)*(1-0.004). Then the probability under the question has the complementary value 1 - (1-0.03)*(1-0.007)*(1-0.004) = 0.0406 = 4.06%. (b) Exactly one of these components will need repair within 1 year. (1-0.03)*(1-0.007)*0.004 + (1-0.03)*0.007*(1-0.004) + 0.03*(1-0.007)*(1-0.004) = 0.04030 = 4.030%. <U>Explanation</U>. The probability that only A-component will need a repair within 1 year is (1-0.03)*(1-0.007)*0.004. The probability that only E-component will need a repair within 1 year is (1-0.03)*0.007*(1-0.004). The probability that only V-component will need a repair within 1 year is 0.03*(1-0.007)*(1-0.004). The probability that exactly one of these three independent events will happen within 1 year is the sum of these particular probabilities, which coincides with my answer. </pre> <H3>Problem 4</H3>A plane has three engines—a central engine and an engine on each wing. The plane will crash only if the central engine fails and at least one of the two wing engines fails. The probability of failure during any given flight is 0.005 for the central engine and 0.008 for each of the wing engines. Assuming that the three engines operate independently, what is the probability that the plane will crash during a flight? <B>Solution</B> <pre> The plane will crash if and only if one of the following events will happen - (a) central and left engines fail - (b) central and right engines fail - (c) central and both left and right engines fail. The probabilities for each of these events are P(a) = 0.005*0.008*(1-0.008) = 0.00003968 P(b) = 0.005*0.008*(1-0.008) = 0.00003968 P(c) = 0.005*0.008*0.008 = 0.00000032 The probability of the crash is the sum P = P(a) + P(b) + P(c) = 0.00007968 (approximately). <U>ANSWER</U> </pre> My other lessons on <B>Probability</B> in this site are <TABLE> <TR> <TD> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Simple-and-simplest-probability-problems.lesson>Simple and simplest probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-probability-problems-using-complementary-probability.lesson>Solving probability problems using complementary probability</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Elementary-Probability-problems-related-to-Combinations.lesson>Elementary Probability problems related to combinations</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-True-False-test.lesson>A True/False test</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-multiple-choice-answers-test.lesson>A multiple choice answers test</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Coinciding-birthdays.lesson>Coinciding birthdays</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-shipment-containing-fair-and-defective-alarm-clocks-.lesson>A shipment containing fair and defective alarm clocks</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/People-in-a-room-write-down-integer-numbers-at-random.lesson>People in a room write down integer numbers at random</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-drawer-contains-a-mixture-of-socks.lesson>A drawer contains a mixture of socks</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Students-studying-foreign-languages.lesson>Students studying foreign languages</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Conditional-probability-problems.lesson>Conditional probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Using-sample-space-to-solve-Probability-problems.lesson>Using sample space to solve Probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Typical-probability-problems-from-the-archive.lesson>Typical probability problems from the archive</A> </TD> <TD> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Geometric-probability-problems.lesson>Geometric probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Advanced-probability-problems-from-the-archive.lesson>Advanced probability problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Challenging-probability-problems.lesson>Challenging probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Selected-probability-problems-from-the-archive.lesson>Selected probability problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Experimental-probability-problems.lesson>Experimental probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Rolling-a-pair-of-fair-dice.lesson>Rolling a pair of fair dice</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Ten-pairs-of-shoes-are-kept-in-a-rack.lesson>Ten pairs of shoes are kept in a rack</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/A-diplomat-and-spies.lesson>A diplomat and spies</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Elementary-operations-on-sets-help-solving-Probability-problems.lesson>Elementary operations on sets help solving Probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Independent-and-mutually-exclusive-events.lesson>Independent and mutually exclusive events</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Unusual-probability-problems.lesson>Unusual probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-problem-for-the-Day-of-April-1.lesson>Probability problem for the Day of April, 1</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/OVERVIEW-of-lessons-on-Probability.lesson>OVERVIEW of lessons on Probability</A> </TD> </TR> </TABLE> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
