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Probability: Die roll until first number

  1. Jul 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Roll a fair die until the first time you roll a '5' then stop. Let X be the number of times you rolled the die. What is the probability that X is divisible by 3?

    2. Relevant equations

    E[X] ?

    3. The attempt at a solution

    I honestly have no idea how to even approach this sort of thing. My first thought was either a bionomial or geometric series, but that could possibly be endless.

    any sort of hints or suggestions would be great.
     
  2. jcsd
  3. Jul 12, 2012 #2

    micromass

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    Let's first start by figuring out the probabilities involved.

    Can you find out [itex]P\{X=1\}[/itex]? (that is: the probability your first throw is a 5). What about [itex]P\{X=2\}[/itex]? (the probability that your first throw is not a five, but your second throw is). Can you find a general formula for [itex]P\{X=n\}[/itex]?
     
  4. Jul 12, 2012 #3
    Find [itex]P(X = x)[/itex]. [itex]X[/itex] is divisible by [itex]3[/itex] if and only if [itex]X = 3k[/itex] for some integer [itex]k[/itex]. Therefore, the probability that [itex]X[/itex] is divisible by [itex]3[/itex] is given by [itex]\sum_{k=1}^{\infty} P(X = 3k)[/itex].
     
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