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What is the expected number of random digits that should be generated to obtain three consecutive zeros?

Now how can i proceed with this problem using the Theorem for Conditioning on Random Variables?

I tried to start with $$E[X]=E[E(X|Y)]$$=$$\sum_{i=1}^\infty E(X|Y=i)P(Y=i)$$

So now, i am thinking how can i proceed using the formula. I know that there are 10 digits and the chance of 1 zero is 1/10 and 9/10 is no zero. But how can i make this into a general formula??

Can anybody explain in words and break down , what the book gives as an answer?

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Mark
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  • related question talking about how many flips to get five consecutive heads. You can use the same method employed by the answers there for your specific problem. – JMoravitz Mar 17 '17 at 21:56
  • @JMoravitz how to use the definition $$E[X]=E[E(X|Y)]$$ – Mark Mar 17 '17 at 22:01
  • Have a look to this answer. http://math.stackexchange.com/questions/1839496/expected-number-of-tosses-to-get-3-consecutive-heads/1839505#1839505 – callculus42 Mar 18 '17 at 06:32
  • @callculus how can i threat E[X|Y] as a random variable. It is still unclear to me. In the examples they dont use the formula – Mark Mar 18 '17 at 15:56

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