Final answer:
The probability that a customer completes a transaction in less than two minutes is approximately 0.3297, or 32.97%.
Step-by-step explanation:
The given problem states that the amount of time it takes for a customer to complete a transaction at a bank follows an exponential distribution with a mean of 5 minutes. We are asked to find the probability that a customer completes a transaction in less than two minutes.
Since the exponential distribution is memoryless, the probability of an event occurring within a given interval is the same regardless of how much time has already passed. Therefore, we can treat the interval between customers as a new exponential distribution with the mean of 5 minutes.
To find the probability that a customer completes a transaction in less than two minutes, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential distribution with mean μ is given by:
CDF(x) = 1 - e^(-x/μ)
Plugging in the values, we have:
CDF(2) = 1 - e^(-2/5) = 1 - e^(-0.4) ≈ 0.3297.
Therefore, the probability that a customer completes a transaction in less than two minutes is approximately 0.3297, or 32.97%.