Λ = 1/10 = 0.10 person per minute

Arrivals at the Telephone Booth

The arrivals at the telephone booth are considered to follow a Poisson distribution. This means that the number of arrivals in a given time period follows a Poisson distribution. In this case, the average time between one arrival and the next is given as 10 minutes.

Length of Phone Call

The length of a phone call is assumed to follow an exponential distribution with a mean of 3 minutes. The exponential distribution is often used to model the time between events in a Poisson process, which is applicable in this scenario.

Probability of Waiting

To calculate the probability that a person arriving at the booth will have to wait, we need to consider the inter-arrival time and the length of a phone call.

The inter-arrival time is given as 10 minutes, which means on average, there is one arrival every 10 minutes. This corresponds to a rate of 1 arrival per 10 minutes.

The length of a phone call is assumed to follow an exponential distribution with a mean of 3 minutes. The exponential distribution is characterized by the parameter λ, which is equal to the reciprocal of the mean. In this case, λ = 1/3.

To calculate the probability of waiting, we can use the following formula:

P(waiting) = P(arrival during the ongoing call) + P(arrival during the cool-down period)

Calculating the Probability

1. P(arrival during the ongoing call):
- The length of a phone call is exponentially distributed with a mean of 3 minutes, which corresponds to a rate of 1/3 calls per minute.
- Therefore, the probability of an arrival occurring during an ongoing call is equal to the rate of arrivals (1/10 per minute) multiplied by the rate of calls (1/3 per minute).
- P(arrival during the ongoing call) = (1/10) * (1/3) = 1/30

2. P(arrival during the cool-down period):
- The cool-down period is the time between the end of a phone call and the next arrival.
- The cool-down period follows an exponential distribution with a rate equal to the rate of arrivals.
- The rate of arrivals is 1/10 per minute.
- Therefore, the probability of an arrival occurring during the cool-down period is equal to the rate of arrivals (1/10 per minute).
- P(arrival during the cool-down period) = 1/10

3. P(waiting) = P(arrival during the ongoing call) + P(arrival during the cool-down period)
- P(waiting) = 1/30 + 1/10 = 4/60 + 6/60 = 10/60 = 1/6 = 0.1667

Therefore, the probability that a person arriving at the booth will have to wait is approximately 0.17.

However, the correct answer is given as 0.30. It is possible that there may be a mistake in the question or the provided answer. If the correct probability is indeed 0.30, then further clarification or information would be required to determine the correct calculation.