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Answer :
Final answer:
The probability of a violent crime occurring in the next minute, given that these crimes follow an exponential distribution and occur every 22 seconds on average, is approximately 91%. Should a violent crime not have occurred in the previous minute, the probability for the next minute remains the same, due to the lack of memory principle in exponential distributions.
Explanation:
The question asks us to determine the probability of a violent crime taking place within a given time frame, making the subject of interest a good example of an exponential distribution. This statistical distribution is often used to model the amount of time until a certain event occurs.
In this case, a violent crime is said to occur every 22 seconds on average. The rate (λ) for the exponential distribution is the reciprocal of this average interval, which equates to approximately 0.045 per second.
Therefore:
- The probability that a violent crime occurs in the next 60 seconds (or one minute) can be calculated using the formula 1 - e^(-λx), where x represents the time in seconds. This leads to a calculation of 1 - e^(-0.045*60), resulting in a probability of around 0.91 or 91%.
- If there has not been a violent crime in the last minute, the likelihood that a crime will occur in the next minute would remain the same. Due to the principle of lack of memory inherent to the exponential distribution, the probability is not influenced by the amount of time already passed without the event happening. So the answer stays 91%.
Learn more about Exponential Distribution here:
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