Suppose that you move to a new house and you are 10% sure that your new house's phone number is 561290. To verify this, you use the house's phone to dial 561290, obtain a busy signal, and conclude that this is indeed your phone number. (Suppose that you were incorrect, and that this is somebody else's phone number. In that case, you will receive a busy signal if and only if that person is making a phone call at the same time. But if you call your own number, you will always receive a busy signal.)

Assume that the probability of a random six-digit phone number being in use at any given time is 0.001. What is the probability that you will be correct in concluding that 561290 is indeed your own phone number? Your answer should be accurate to 3 decimal places.

In order to find the probability that you will be correct in concluding that 561290 is your own phone number, we need to consider two scenarios:

Scenario 1: 561290 is your own phone number.
Scenario 2: 561290 is somebody else's phone number.

Let's break down these scenarios and calculate the probabilities:

Scenario 1:
The probability that 561290 is your own phone number is given as 10%. Therefore, the probability of this scenario is 0.10.

Scenario 2:
The probability that 561290 is somebody else's phone number is given as 90% (since it is not your phone number). Therefore, the probability of this scenario is 0.90.

Now we need to calculate the probabilities of getting a busy signal in each scenario.

Scenario 1:
If 561290 is your own phone number, you will always receive a busy signal when calling it. Therefore, the probability of getting a busy signal in this scenario is 1.00.

Scenario 2:
If 561290 is somebody else's phone number, you will receive a busy signal only if that person is making a phone call at the same time. The probability that a random six-digit phone number is in use at any given time is given as 0.001. Therefore, the probability of getting a busy signal in this scenario is 0.001.

Now we can calculate the overall probability that you will be correct in concluding that 561290 is indeed your own phone number by using the Law of Total Probability:

P(Correct conclusion) = P(Scenario 1) * P(Busy signal in Scenario 1) + P(Scenario 2) * P(Busy signal in Scenario 2)
= (0.10 * 1.00) + (0.90 * 0.001)
= 0.10 + 0.0009
= 0.1009

Therefore, the probability that you will be correct in concluding that 561290 is your own phone number is approximately 0.101 (accurate to 3 decimal places).