Question

A home security system is designed to have a \(99 \%\) reliability rate. Suppose that nine homes equipped with this system experience an attempted burglary. Find the probabilities of these events: a. At least one of the alarms is triggered. b. More than seven of the alarms are triggered. c. Eight or fewer alarms are triggered.

Short answer

Question: In a neighborhood with nine homes, there is a 99% chance that a burglar alarm will be triggered in the event of a break-in. What are the probabilities of the following: a. At least one alarm is triggered. b. More than seven alarms are triggered. c. Eight or fewer alarms are triggered. Answer: a. The probability of at least one alarm being triggered is approximately 0.9999999996. b. The probability of more than seven alarms being triggered is approximately 0.7988. c. The probability of eight or fewer alarms being triggered is approximately 0.0865.

Step-by-step solution

Define parameters

Let n = 9 (number of homes), p = 0.99 (probability of success, i.e., alarm triggered), and q = 1 - p = 0.01 (probability of failure, i.e., alarm not triggered).

Calculate probabilities#for_at_least_one_success

To find the probability of at least one alarm being triggered, we calculate the complementary probability of no alarms being triggered: 1 - P(X=0) = 1 - (C(9,0) * p^0 * q^9) = 1 - (1 * 0.99^0 * 0.01^9) ≈ 1 - 0.0000000003874 ≈ 0.9999999996

Calculate probabilities for more_than_seven_successes

To find the probability of more than seven alarms being triggered, we find the sum of probabilities of exactly eight and exactly nine alarms being triggered: P(X>7) = P(X=8) + P(X=9) = C(9,8) * p^8 * q^1 + C(9,9) * p^9 * q^0 ≈ 0.387420487 * 0.99^8 * 0.01 + 1 * 0.99^9 * 0.01^0 ≈ 0.7988

Calculate probabilities for eight_or_fewer_successes

To find the probability of eight or fewer alarms being triggered, we calculate the complementary probability of exactly nine alarms being triggered: 1 - P(X=9) = 1 - (C(9,9) * p^9 * q^0) = 1 - (1 * 0.99^9 * 0.01^0) ≈ 1 - 0.9135 ≈ 0.0865

Present the results

a. The probability of at least one alarm being triggered is approximately 0.9999999996. b. The probability of more than seven alarms being triggered is approximately 0.7988. c. The probability of eight or fewer alarms being triggered is approximately 0.0865.

Key concepts

Reliability in Probability

Reliability in probability refers to the likelihood that a system performs its intended function without failure. In this exercise, the security system has a reliability of 99%. This means that each alarm is 99% likely to trigger when needed. High reliability is crucial in systems like security alarms because failures can lead to significant consequences.

Reliability of a system is often expressed as a percentage:

• This tells us how dependable the system is.
• For 9 homes, each alarm has the same probability (99%) of going off.
• Calculating events such as 'at least one alarm triggered' involves understanding this reliability.

When working with probabilities, remember that the reliability percentage is the starting point for all calculations. It defines the success rate and can be adjusted to understand probabilities of multiple events.

Understanding Complementary Probability

Complementary probability is a fundamental concept especially useful for finding the probability of 'at least one' event happening. The complement of an event is the probability that the event does not occur.

In this exercise:

• The complementary probability is calculated by evaluating the scenario where no alarms are triggered.
• The probability that at least one alarm is triggered can be found using: \[1 - P(X=0)\]
• This computes the complement of no alarm triggering, giving us the probability of at least one event occurring.

Using complementary probability simplifies problems, avoids long calculations and is essential for efficiently determining the occurrence of events within a larger sample space.

Using the Combination Formula

The combination formula is used to determine the number of ways to choose items from a larger set without considering the order. In probability, it's crucial for calculating events where order doesn't matter, like how many alarms are triggered.

The formula is:\[C(n, k) = \frac{n!}{k!(n-k)!}\]

• Here, \(n\) is the total number of items, and \(k\) is the number of items to choose.
• It helps find probabilities like exactly eight or nine alarms being triggered in the exercise.
• Calculations, such as \(C(9, 8)\) or \(C(9, 9)\), determine how many combinations exist for these specific numbers of successes.

Understanding this concept is vital because it enables precise calculation of probabilities where multiple outcomes are possible. It ensures you consider all potential scenarios without bias from sequence or arrangement.