4. Probability
Multiplication Rule: Independent Events
2:01 minutesProblem 4.2.26a
Textbook Question
Alarm Clock Life Hack Each of us must sometimes wake up early for something really important, such as a final exam, job interview, or an early flight. (Professional golfer Jim Furyk was disqualified from a tournament when his cellphone lost power and he overslept.) Assume that a battery-powered alarm clock has a 0.005 probability of failure, a smartphone alarm clock has a 0.052 probability of failure, and an electric alarm clock has a 0.001 probability of failure.
a. What is the probability that your single battery-powered alarm clock works successfully when you need it?
Verified step by step guidance1
Step 1: Understand the problem. The probability of failure for the battery-powered alarm clock is given as 0.005. To find the probability of success, we need to calculate the complement of the failure probability.
Step 2: Recall the complement rule in probability. The complement rule states that the probability of an event not occurring (failure) plus the probability of the event occurring (success) equals 1. Mathematically, this is expressed as: P(Success) = 1 - P(Failure).
Step 3: Substitute the given probability of failure into the complement formula. Here, P(Failure) = 0.005, so the formula becomes: P(Success) = 1 - 0.005.
Step 4: Perform the subtraction to find the probability of success. This step involves subtracting the failure probability from 1.
Step 5: Interpret the result. The calculated probability represents the likelihood that the battery-powered alarm clock will work successfully when needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. A probability of 0 indicates that the event will not happen, while a probability of 1 indicates certainty. In this context, the probabilities given for each type of alarm clock represent the chance of failure, which can be used to calculate the chance of success.
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Complementary Events
Complementary events are pairs of outcomes where one event occurs if and only if the other does not. For example, if the probability of an alarm clock failing is 0.005, the probability of it working successfully is the complement, calculated as 1 minus the failure probability. Understanding complementary events is crucial for determining the likelihood of success in scenarios involving failure rates.
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Independent Events
Independent events are those whose outcomes do not affect each other. In the context of alarm clocks, if you consider multiple clocks, the failure of one does not influence the others. This concept is important when calculating the overall probability of success or failure when using multiple devices, as it allows for straightforward multiplication of individual probabilities.
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