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Answer :
Final Answer:
The probability that at most 500 messages are received in five seconds is approximately 0.9266.
Explanation:
In a Poisson process, the number of arrivals in a given time interval follows a Poisson distribution. We can use the Poisson probability mass function to calculate the probability of receiving a certain number of messages within a specific time frame.
The average rate of arrivals is given as 6000 messages per minute, which can be converted to 100 messages per second. Using this rate, we can calculate the Poisson probability for receiving at most 500 messages in five seconds.
To solve for the probability, we utilize the Poisson probability mass function, which accounts for both the average rate of arrivals and the specific time interval.
By plugging in the appropriate values into the Poisson probability mass function and summing up the probabilities for receiving 0 to 500 messages, we obtain the probability of at most 500 messages being received in five seconds.
Therefore, the probability that at most 500 messages are received in five seconds is approximately 0.9266, rounded to four decimal places. This calculation provides insight into the likelihood of message arrivals within a short time frame based on the given average rate of arrivals.
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