Probability Problem for Phone Numbers.

[plumlogan]

This is a challange. I'm bad at math so I can't meet it, but I thought someone might be interested in tackling it.

What is the method and solution for the following problem:

Given the ten digits on a normal phone, what is the probability that a random ten digit number (such as a phone number w/the area code included) will not contain three consecutive integers, i.e. 3,4,5 or 8,9,0 or 1,2,3?

This question has been asked arbitrarily around the office and nobody has any idea where to start.

 

[galactus]

Assuming the first digit can not be a 0 or 1 and repetition is allowed, there are \(\displaystyle \L\\(8)(8)\cdot{10^{8}}=6,400,000,000\) possible combinations.

Incidentally, 8,9,0 are not consecutive numbers.

There are 8 possible sets of 3 consecutive integers:

0,1,2; 1,2,3; 2,3,4; 3,4,5; 4,5,6; 5,6,7; 6,7,8; 7,8,9

The thing you could do is bundle up a group of 3 consecutives and place them in one of 8 places in the string. The first two (0,1,2 and 1,2,3) are not allowed at the beginning because of the 0 and 1.

Let's place one of them and count it as 8 integers to arrange in 8!=40320 ways. There are 6 sets we can place throughout in 8 different places.
48*8!=1935360

The 0,1,2 and 1,2,3 we can place in 7 places each, so 14*8!=564480.

There are 2499840 ways to place the 8 sets of 3 consecutive integers.

The probability of having three consecutives is \(\displaystyle \L\\\frac{62\cdot{8!}}{64\cdot{10^{8}}}=\frac{1953}{5000000}\approx{0.0003906}\)

Therefore, the probability that there are no consecutives is \(\displaystyle \L\\\frac{4998047}{5000000}\approx{0.9996094}\)

I could have easily over or undercounted. Perhaps someone will be along to confirm or deny my results.

 

[pka]

Incidentally, 8,9,0 are not consecutive numbers.

I can see why someone constructing this problem might include 8,9,0 as a consecutive triple: 8,9,10.
If that is specified then there are eight sequences we want to exclude.
Using inclusion/exclusion how many ways are there to include at least one of those triple?
Then do the subtraction.