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Math Help - Probability of a probability?

  1. #1
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    Probability of a probability?

    Hi, I'm just doing some personal problems and I ran into this thought...

    Take for instance this passphrase: "dog.cat.mouse"
    (which is inclusive of the periods but exclusive of the quotations, so a total of 13 characters)

    The goal is to guess this passphrase, using the set of American English alphabet, both lowercase and uppercase, positive numbers 0-9, and the following symbols: [ ] \ ; ' / , .
    So that's a total of 70 characters.

    A total of 30,000,000,000 (thirty billion) guesses are possible every 1 second. The length of the passphrase is unknown.

    The question: what is the probability that this passphrase will be guessed within the time period of 500 years?

    The only way I could think of approaching this problem would be to find the probability of guessing the passphrase within a second, then calculate the number of seconds in 500 years, and then multiply the previously aforementioned probability with the probability of guessing it in 500 years? I'm not sure...

    How would I go about this?
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  2. #2
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    Re: Probability of a probability?

    Hello, daigo!

    Take for instance this passphrase: "dog.cat.mouse" (which is inclusive
    of the periods but exclusive of the quotations, so a total of 13 characters)

    The goal is to guess this passphrase, using the set of American English alphabet,
    both lowercase and uppercase, positive numbers 0-9, and the following symbols:
    [ ] \ ; ' / , . . . So that's a total of 70 characters.

    A total of 30,000,000,000 (thirty billion) guesses are possible every 1 second.
    The length of the passphrase is unknown.

    What is the probability that this passphrase will be guessed within 500 years?

    If the passphrase is n characters long, there are 70^n possible passphrases.

    How many guesses can be made in 500 years?

    \text{1 minute } \,=\,\text{ 60 seconds}

    \text{1 hour }\,=\,\text{ 60 minutes }\,=\,\text{ 3600 seconds}

    \text{1 day }\,=\,\text{ 24 hours }\,=\,\text{ 86,400 seconds}

    \text{1 year }\,=\;365\tfrac{1}{4}\text{ days }\,=\,\text{ 31,192,350 seconds}

    \text{500 years }\,=\;15,\!596,\!175,\!000\;\approx\;1.56\times10^  {10}\text{ seconds}


    \text{Hence: }\:(30\times 10^9) \times (1.56\times 10^{10}) \:=\:4.68 \times 10^{20}\text{ guesses in 500 years.}


    \text{The probability is: }\:\frac{4.68 \times 10^{20}}{70^n}

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