# Thread: Probability of a probability?

1. ## 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...

2. ## 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 $\displaystyle n$ characters long, there are $\displaystyle 70^n$ possible passphrases.

How many guesses can be made in 500 years?

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

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

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

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

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

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

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