past history indicates that an average of 1.2 calls are received per minute. What is the probability of 3 or more calls per minute
If the distribution is normal, and the std is high enough, the probability could approach 0.5
To calculate the probability of 3 or more calls per minute, we need to use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space when the average rate of occurrence is known.
The formula for the Poisson probability is as follows:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of k events occurring
- e is the base of the natural logarithm (approximately 2.71828)
- λ (lambda) is the average rate of occurrence of the events
- k is the number of events we're interested in
In this case, the average rate of occurrence is given as 1.2 calls per minute. We want to calculate the probability of 3 or more calls per minute, so we need to calculate the probabilities for 3, 4, 5, and so on.
Let's calculate the probability of 3 or more calls per minute:
P(X ≥ 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2)
To calculate P(X = k), we substitute the values into the Poisson probability formula. For example, P(X = 0) would be:
P(X = 0) = (e^(-1.2) * 1.2^0) / 0!
Since any number raised to the power of 0 is 1, and the factorial of 0 is 1, we can simplify this to:
P(X = 0) = e^(-1.2)
Similarly, we can calculate P(X = 1) and P(X = 2) using the same formula.
Finally, we subtract the sum of these probabilities from 1 to get the probability of 3 or more calls per minute.
Note: Remember to use the appropriate scientific calculator or statistical software to calculate the exponential function (e^x) accurately.