What is the probability that a 6-digit phone number contains at least one 2? (Repetition of numbers and lead zero are allowed).

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- Jan 7th 2009, 04:23 PMasugirl87probability
What is the probability that a 6-digit phone number contains at least one 2? (Repetition of numbers and lead zero are allowed).

- Jan 7th 2009, 04:38 PMMush
Imagine the situation as being a room with 6 phones. Each phone has 10 digits on it, and you have to pick one digit from each phone!

As such there are 10 different ways you can choose 1 from 10. And you have to choose 6 of these. Hence there are $\displaystyle 10^6$ combinations!

Now lets consider the same situation, only now all the 2s on the phones have been removed. So now there are 6 phones, each with only 9 digits, (0,1,3,4,5,6,7,8,9). So you are choosing 1 button from 9, 6 times! Hence $\displaystyle 9^6$.

So we have concluded that there are $\displaystyle 10^6 $ combinations, and $\displaystyle 9^6 $ of them have no 2!

Hence the probability that there is at least one 2 is :

$\displaystyle 10^6 - 9^6 = 468559$ - Jan 29th 2009, 07:39 AMramvenk
The response is attached as a gif image, hope this helps you.