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Chapter 5: Problem 24

A large bank vault has several automatic burglar alarms. The probability is 0.55 that a single alarm will detect a burglar. (a) How many such alarms should be used for \(99 \%\) certainty that a burglar trying to enter will be detected by at least one alarm? (b) Suppose the bank installs nine alarms. What is the expected number of alarms that will detect a burglar?

Short Answer

Expert verified
(a) The bank needs 6 alarms; (b) Expected value is 4.95 alarms.

Step by step solution

01

Understand the Problem

We need to find the number of alarms required such that the probability of at least one detecting a burglar is 99%. Each alarm independently detects a burglar with a probability of 0.55, and does not detect a burglar with a probability of 0.45.
02

Set Up the Probability Equation

The probability that one alarm will not detect a burglar is 0.45. Therefore, if there are \( n \) alarms, the probability that none of them detects a burglar is \( 0.45^n \). We need this probability to be less than 0.01 (since we want a 99% certainty of detection).
03

Solve the Inequality for Part (a)

Set up the inequality: \( 0.45^n < 0.01 \). Taking the logarithm on both sides, we get: \( n \cdot \log(0.45) < \log(0.01) \). Solving for \( n \): \[ n > \frac{\log(0.01)}{\log(0.45)} \approx \frac{-2}{-0.3468} \approx 5.77 \]. Thus, \( n \) must be at least 6.
04

Expectation Calculation for Part (b)

For part (b), we need to calculate the expected number of alarms detecting a burglar. This involves finding the expected value of a binomial distribution with parameters \( n = 9 \) and \( p = 0.55 \). The expected value is calculated as \( E[X] = n \times p = 9 \times 0.55 = 4.95 \).
05

Conclusion

For part (a), the bank needs at least 6 alarms for 99% certainty. For part (b), with 9 alarms, the expected number of alarms detecting a burglar is 4.95.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binomial Distribution
The concept of binomial distribution is essential when dealing with experiments or scenarios where we have a fixed number of independent trials, each with two possible outcomes: success or failure. In the context of burglar alarms, each alarm is seen as a trial. The alarm can either detect the burglar (success) or not detect them (failure).

Key features of a binomial distribution include:
  • Fixed number of trials (denoted by \( n \)).
  • Each trial is independent of the others.
  • The probability of success is constant for each trial (denoted by \( p \)).
For our bank vault problem, the binomial distribution helps us determine the likelihood of different numbers of alarms detecting a burglar, given each alarm detects with a probability of 0.55.
Calculating Expected Value
Expected value is a critical concept in probability that helps us understand the average outcome we can expect given a probability distribution. Specifically, for a binomial distribution, the expected value is calculated using the formula: \[ E[X] = n \times p \] where \( n \) is the number of trials, and \( p \) is the probability of success on each trial.

In our scenario with nine alarms, each with a detection probability of 0.55, the expected number of alarms detecting the burglar is: \[ E[X] = 9 \times 0.55 = 4.95 \] This means on average, about 4.95 alarms (you can think of this as approximately 5 alarms in practical terms) will detect a burglar when they try to enter.
Understanding Logarithms in Probability
Logarithms are a powerful mathematical tool used to deal with equations involving exponential growth or decay. They are especially useful in probability when solving for the number of trials needed for a certain probability threshold, like in our problem where we want the probability of detection to be at least 99%.

For such problems, we often end up with equations like \( a^n = b \). To solve for \( n \), we take the logarithm of both sides, which converts the equation into a linear one: \[ n \times \log(a) = \log(b) \] This transformation helps solve for \( n \) by isolating it on one side of the equation. In our vault example, we applied logarithms to find the number of alarms needed to achieve the 99% certainty of detection.
Concept of Independent Events
In probability, the concept of independent events refers to situations where the outcome of one event does not influence the outcome of another. Each alarm in our problem detects a burglar independently of the others.

Here鈥檚 what defines independent events:
  • The probability of one event occurring does not affect the probability of another event.
  • The combined probability of independent events is the product of their individual probabilities.
For the bank鈥檚 alarms, this means each alarm has its own chance (0.55) of detecting a burglar, and this chance remains unchanged regardless of the outcome of other alarms. Understanding the independence of these events is crucial to accurately calculating probabilities and expectations in situations with multiple alarms.

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Most popular questions from this chapter

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Much of Trail Ridge Road in Rocky Mountain National Park is over 12,000 feet high. Although it is a beautiful drive in summer months, in winter the road is closed because of severe weather conditions. Winter Wind Studies in Rocky Mountain National Park by Glidden (published by Rocky Mountain Nature Association) states that sustained galeforce winds (over 32 miles per hour and often over 90 miles per hour) occur on the average of once every 60 hours at a Trail Ridge Road weather station. (a) Let \(r=\) frequency with which gale-force winds occur in a given time interval. Explain why the Poisson probability distribution would be a good choice for the random variable \(r\) (b) For an interval of 108 hours, what are the probabilities that \(r=2,3,\) and 4? What is the probability that \(r<2 ?\) (c) For an interval of 180 hours, what are the probabilities that \(r=3,4,\) and 5? What is the probability that \(r<3 ?\)
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