Probability...?

Question

Joe will roll a pair of dice until he rolls a 7 (sum of 2 numbers). Let X be number of times he will roll without getting a 7.
(For example, if on the third roll Joe first rolls 7, then X=2.)
What is the expected value of X?

[hints:
let p(x) denote probability that X=x.
The expected value will be an infinite series, (0)p(0) + (1)p(1) + (2)p(2) + ... .
After the formula for p(n) is determined, compute the series (remember 1 + x + x^2 + x^3 + ... = 1/(1-x), if |x|<1)
Take the derivative of both sides.
]

yeah yeah, you got your 2 points...

Answer 1

Calculate the expected number of rolls for the first seven and subtract one.

S = 1/6 + (5/6)(2/6) + (5/6)²(3/6) + (5/6)³(4/6) + ...
(5/6)S = (5/6)(1/6) + (5/6)²(2/6) + (5/6)³(3/6) + ...

Subtract the second equation from the first.

(1/6)S = 1/6 + (5/6)(1/6) + (5/6)²(1/6) + (5/6)³(1/6) + ...

Multiply by 6.

S = 1 + (5/6) + (5/6)² + (5/6)³ + ...
(5/6)S = (5/6) + (5/6)² + (5/6)³ + ...

Subtract the second equation from the first.

(1/6)S = 1
S = 6

The expected number of rolls for the first seven is six.

So X = 6 - 1 = 5

Answer 2

That's all very well Miss Shazaam but what is the price of eggs this week?

Answer 3

You are a nice girl, for giving points away like this... Thanks!!!