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Chapter 23: Problem 4

A machine makes resistors of which \(96 \%\) are acceptable and \(4 \%\) are unacceptable. Three resistors are picked at random. Calculate the probability that (a) all are acceptable (b) all are unacceptable (c) at least one is unacceptable.

Short Answer

Expert verified
Answer: The probability of at least one resistor being unacceptable is 0.115264.

Step by step solution

01

Understand the given probabilities

The probability of a resistor being acceptable is \(96\%\), which means the probability is \(0.96\). Similarly, the probability of a resistor being unacceptable is \(4\%\), meaning the probability is \(0.04\). We will use these probabilities to calculate the desired probabilities in the given scenarios.
02

Calculate the probability of all resistors being acceptable (a)

Since we are choosing the resistors independently, we can simply multiply the probability of each resistor being acceptable. P(all are acceptable) = P(1st is acceptable) * P(2nd is acceptable) * P(3rd is acceptable) P(all are acceptable) = \(0.96 * 0.96 * 0.96 = 0.884736\)
03

Calculate the probability of all resistors being unacceptable (b)

Similarly, for all the resistors being unacceptable, we just have to multiply the probability of each resistor being unacceptable. P(all are unacceptable) = P(1st is unacceptable) * P(2nd is unacceptable) * P(3rd is unacceptable) P(all are unacceptable) = \(0.04 * 0.04 * 0.04 = 0.000064\)
04

Calculate the probability of at least one resistor being unacceptable (c)

To find the probability of having at least one unacceptable resistor, we can use the complementary probability. The complementary event of having at least one unacceptable resistor is having all resistors acceptable. We have already computed the probability of all resistors being acceptable in part (a). P(at least one is unacceptable) = 1 - P(all are acceptable) P(at least one is unacceptable) = \(1 - 0.884736 = 0.115264\) Now, we have calculated the probabilities for each scenario: (a) The probability of all resistors being acceptable is \(0.884736\). (b) The probability of all resistors being unacceptable is \(0.000064\). (c) The probability of at least one resistor being unacceptable is \(0.115264\).

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Most popular questions from this chapter

The resistances of 50 resistors are measured and the results recorded as follows: $$ \begin{array}{cc} \hline \text { Resistance }(\Omega) & \text { Frequency } \\ \hline 5.0 & 17 \\ 5.5 & 12 \\ 6.0 & 10 \\ 6.5 & 6 \\ 7.0 & 5 \\ \hline \end{array} $$ Calculate the standard deviation of the measurements.

A fair die is rolled. Calculate the probability that the score is (a) 4 (b) 4 or more (c) more than 4 (d) not 4

A service engineer receives on average seven calls in a 24-hour period. Calculate the probability that in a 24 -hour period the engineer receives (a) seven calls (b) eight calls (c) six calls (d) fewer than three calls

The probability of passing a module on the first attempt is \(0.9\). A student takes six modules. Calculate the probability that the student (a) passes five modules (b) passes all modules (c) is required to take two or more resits.

The probability that a component works is \(0.92\). An engineer wants to be at least \(99 \%\) certain of carrying six working components. Calculate the minimum number of components that the engineer needs to carry.
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