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Chapter 23: Problem 2

The probability that a component fails within a month is \(0.009\). If 800 components are examined calculate the probability that the number failing within a month is (a) nine, (b) five, (c) less than three, (d) four or more.

Short Answer

Expert verified
Based on the given probability of a single component failing within a month, the probabilities for different outcomes for 800 components are: (a) The probability of exactly 9 failed components is approximately 0.170 (17%). (b) The probability of exactly 5 failed components is approximately 0.007 (0.7%). (c) The probability of less than 3 failed components is approximately 0.0002 (0.02%). (d) The probability of 4 or more failed components is approximately 0.9998 (99.98%).

Step by step solution

01

Identify given information

In this problem, we are given the number of components (n = 800) and the probability of a component failing within a month (p = 0.009).
02

Calculate the probability for 9 failed components

Using the binomial probability formula, we can find the probability for having exactly 9 failed components (k = 9): P(X = 9) = C(800, 9) * (0.009)^9 * (1 - 0.009)^(800 - 9) Using a calculator or software, we get: P(X = 9) ≈ 0.170
03

Calculate the probability for 5 failed components

Now, let's find the probability for having exactly 5 failed components (k = 5): P(X = 5) = C(800, 5) * (0.009)^5 * (1 - 0.009)^(800 - 5) Using a calculator or software, we get: P(X = 5) ≈ 0.007
04

Calculate the probability for less than 3 failed components

To find the probability for having less than 3 failed components, we need to find P(X = 0), P(X = 1), and P(X = 2) and add them together: P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = [C(800, 0) * (0.009)^0 * (1 - 0.009)^(800 - 0)] + [C(800, 1) * (0.009)^1 * (1 - 0.009)^(800 - 1)]+ [C(800, 2) * (0.009)^2 * (1 - 0.009)^(800 - 2)] Using a calculator or software, we get: P(X < 3) ≈ 0.0002
05

Calculate the probability for 4 or more failed components

Since the total probability of all events is 1, to find the probability of having 4 or more failed components, we can find the complement probability (subtract the probabilities of having 0 to 3 failed components from 1): P(X ≥ 4) = 1 - P(X < 4) = [1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3))] Using the same method as in Step 4, we can calculate the probabilities for 0 to 3 failed components and then subtract their sum from 1: P(X ≥ 4) ≈ 1 - (0.0002) ≈ 0.9998 So, the probabilities for the different scenarios are: (a) P(X = 9) ≈ 0.170 (b) P(X = 5) ≈ 0.007 (c) P(X < 3) ≈ 0.0002 (d) P(X ≥ 4) ≈ 0.9998

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Most popular questions from this chapter

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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