SOLUTION: We are rolling a 6 sided die. Let A be the random event of how many rolls it takes to roll your first 4. What is the probability mass function and prove that the values of f(x) a

Algebra ->  Probability-and-statistics -> SOLUTION: We are rolling a 6 sided die. Let A be the random event of how many rolls it takes to roll your first 4. What is the probability mass function and prove that the values of f(x) a      Log On


   

Question 1166511: We are rolling a 6 sided die. Let A be the random event of how many rolls it takes to roll your first 4. What is the probability mass function and prove that the values of f(x) add up to 1.
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
The probability that the 1st roll is a 4 is 
1%2F6, the 1st term
The probability that the 1st roll is not a 4 and the 2nd one is a 4 is
%285%2F6%29%2A%281%2F6%29, the 2nd term
The probability that the 1st 2 rolls are not 4's and the 3rd one is a 4 is
%285%2F6%29%285%2F6%29%2A%281%2F6%29=%285%2F6%29%5E2%2A%281%2F6%29, the 3rd term
The probability that the 1st 3 rolls are not 4's and the 4th one is a 4 is
%285%2F6%29%285%2F6%29%285%2F6%29%2A%281%2F6%29=%285%2F6%29%5E3%2A%281%2F6%29, the 4th term
...
The probability that the 1st x-1 rolls are not 4's and the xth one is a 4 is
%285%2F6%29%5E%28x-1%29%2A%281%2F6%29, the xth term

So

f%28x%29=sum%28%28%285%2F6%29%5E%28n-1%29%2A%281%2F6%29%29%2Cn=1%2Cx%29

Notice that theoretically (mathematically) it could take you a billion
rolls, or a million years, [lol] to roll your first 4. [if you had an
extremely stubborn die!!!]

The sum of all values of f(x) is an infinite series and the sum of the
infinite series is:

s%5Binfinity%5D=a%5B1%5D%2F%281-r%5E%22%22%29

where a1 = first term = 1/6, and r = 5/6



Edwin