Final answer:
To find the probability of receiving at least 5 messages in the next hour, we can use the Poisson distribution and the complement rule. The average number of messages per hour is 9, so we can calculate the probabilities of receiving 0, 1, 2, 3, and 4 messages using the Poisson distribution formula. By summing up these probabilities, we can find the probability of receiving less than 5 messages. Finally, we can subtract this probability from 1 to find the probability of receiving at least 5 messages.
Step-by-step explanation:
To find the probability of receiving at least 5 messages in the next hour, we need to use the Poisson distribution. The average number of messages per hour is 9, so we can use this as the lambda value for the Poisson distribution. The formula for the probability mass function of the Poisson distribution is P(X = x) = (e^(-lambda) * lambda^x) / x!, where X is the random variable representing the number of messages received in an hour, lambda is the average number of messages per hour, and x is the number of messages we're interested in.
To find the probability of receiving at least 5 messages, we can sum up the probabilities of receiving 5, 6, 7, 8, 9, 10, and so on, up to infinity. However, since summing up infinite terms is not practical, we can use the complement rule to find the probability of receiving less than 5 messages and subtract it from 1 to find the probability of receiving at least 5 messages.
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
We can use the formula for the Poisson distribution to calculate the probabilities:
P(X = x) = (e^(-lambda) * lambda^x) / x!
Using the average of 9 messages per hour as the lambda value, we can plug in the values of x into the formula and calculate the probabilities for each value:
P(X = 0) = (e^(-9) * 9^0) / 0! = (e^(-9) * 1) / 1
P(X = 1) = (e^(-9) * 9^1) / 1! = (e^(-9) * 9) / 1
P(X = 2) = (e^(-9) * 9^2) / 2! = (e^(-9) * 81) / 2
P(X = 3) = (e^(-9) * 9^3) / 3! = (e^(-9) * 729) / 6
P(X = 4) = (e^(-9) * 9^4) / 4! = (e^(-9) * 6561) / 24
Now we can sum up these probabilities to find P(X < 5):
P(X < 5) = (e^(-9) * 1) / 1 + (e^(-9) * 9) / 1 + (e^(-9) * 81) / 2 + (e^(-9) * 729) / 6 + (e^(-9) * 6561) / 24
Finally, we can use the complement rule to find the probability of receiving at least 5 messages:
P(X >= 5) = 1 - P(X < 5)