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.

3 answers

Bayes rule

0.991

0.1/(0.1+0.9*0.001)

To solve this problem, we need to calculate the conditional probability that the number 561290 is the correct phone number given that we received a busy signal when we dialed it.

Let's break down the problem into the relevant probabilities:

P(Correct Number) = 0.001 (probability of a random six-digit number being in use)
P(Incorrect Number) = 1 - P(Correct Number) = 1 - 0.001 = 0.999

P(Busy Signal | Correct Number) = 1 (we always get a busy signal when we dial our own number)
P(Busy Signal | Incorrect Number) = 0.001 (the probability of getting a busy signal when dialing any other number)

Now, using Bayes' theorem, we can calculate the conditional probability:

P(Correct Number | Busy Signal) = (P(Busy Signal | Correct Number) * P(Correct Number)) / P(Busy Signal)

We need to find P(Busy Signal) by considering both possibilities of getting a busy signal:

P(Busy Signal) = P(Busy Signal | Correct Number) * P(Correct Number) + P(Busy Signal | Incorrect Number) * P(Incorrect Number)
= 1 * 0.001 + 0.001 * 0.999
= 0.001 + 0.000999
= 0.001999

Now we can calculate the desired probability:

P(Correct Number | Busy Signal) = (1 * 0.001) / 0.001999
= 0.001 / 0.001999
= 0.50025

Therefore, the probability of being correct in concluding that 561290 is your own phone number is approximately 0.500 (accurate to 3 decimal places).