Expert: Paul Klarreich - 1/25/2008

Question
Hi,   
Would you please help me with following question:   
Given two independent, fair 6-sided dice:
(1) What is the probability that X1 = 1 and X2 = 1?
(2) What is the probability that X1 = 1 or X2 = 1?
(3) What is the expected value of the sum X1 + X2?
(4) Is the answer the same when they are not independent?  (5) What is the expected number of times needed to throw one die to achieve a 6 at least once?

Answer
Questioner:   Sandy
Category:  Calculus
Private:  No
 
Subject:  Dice
Question:  Hi,   
Would you please help me with following question:   
Given two independent, fair 6-sided dice:
(1) What is the probability that X1 = 1 and X2 = 1?
(2) What is the probability that X1 = 1 or X2 = 1?
(3) What is the expected value of the sum X1 + X2?
(4) Is the answer the same when they are not independent?  (5) What is the expected number of times needed to throw one die to achieve a 6 at least once?
.................................................

1. Independent means p(A B) = p(A) p(B) = 1/6 * 1/6 = 1/36

2. p(A or B) = p(A) + p(B) - P(A B) = 1/6 + 1/6 - 1/36 = 11/36

3. Add these products:

  1 * 2
  2 * 3
  3 * 4
  4 * 5
  5 * 6
  6 * 7
  5 * 8
  4 * 9
  3 * 10
  2 * 11
  1 * 12

Then divide by 36.  The answer should be 7.


(4) Is the answer the same when they are not independent?  

--- What do you mean by THEY?


(5) What is the expected number of times needed to throw one die to achieve a 6 at least once?

For this, you must compute:

p(k) means probability of 'no 6 on rolls 1..k-1, and 6 on roll k.'

p(k) = (5/6)^k-1 (1/6)

Now you want the total of  k * p(k) -- that's the expectation.
inf
SUM k (5/6)^k-1 (1/6)
k=1

inf
SUM k 5^k-1/6^k
k=1

That converges slowly, so you'll have to do some arithmetic.  (Excel is good for this.)