Question

I am trying to figure out the probability, expected value, variance, and standard deviation for a series of dice rolls. For example, if I roll a six-sided die in an attempt to roll a 1, and it takes me 7 rolls before a 1 appears, what are those answers? I have figured out the probability equation:

P(P-1)^x where x is the number of rolls - 1 so for 7 rolls the probability would be: 1/6(1-1/6)^6 = 0.05581632...

Further where I am lost is taking the above and using it to find the Expected Value, Variance, and Standard Deviation?

As I see the equations but plugging in numbers has me lost as p is the probability of failure and x = 0,1,2,3 for geometric distribution it would be

E(X)= (1-p)/p .... this is where I am lost as failure is 5/6 not 1/6 correct? Please show example of this so I can better understand, also on Variance, and Standard Deviation?

Answer 1

If I roll a six-sided die in an attempt to roll a 1, and it takes me 7 rolls before a 1 appears, it is a Geometric Distribution as dscribed below:

Geometric Distribution:

The probability of success of xth trial is given by:

,

                             for x = 1,2,...

Here,

p = 1/6

q = 5/6

x = 7

Substituting, we get:

Expected Value = E(X) for Geometric Distribution is:

Substituting p = 1/6, we get:

Variance Var(X) for Geometric Distribution is:

Substituting p = 1/6 and q = 5/6, we get:

Standard Deviation =