show the sample space when 2 unbiased dice are thrown also compare the probabilities of all possible sum of outcomes appear. Also prepare a detailed note including examples that why the probability of each possible sum of dice is not same.
Answers
Given : 2 unbiased dice are thrown
To Find : sample space
probabilities of all possible sum of outcomes appear.
Solution:
unbiased dice has numbers from 1 to 6
two Dice are rolled hence total output = 6 * 6 = 36
hence n(S) = 36
Numbers of elements in the sample space 36
S = { (1 , 1) , ( 1 , 2) , ( 1 , 3) , ( 1, 4) , ( 1, 5) , ( 1 , 6) ,
(2 , 1) , ( 2 , 2) , ( 2 , 3) , ( 2, 4) , ( 2, 5) , ( 2 , 6) ,
(3 , 1) , ( 3 , 2) , (3 , 3) , (3, 4) , ( 3, 5) , ( 3 , 6) ,
(4 , 1) , ( 4 , 2) , (4 , 3) , ( 4, 4) , (4, 5) , ( 4, 6) ,
(5, 1) , ( 5 , 2) , ( 5 , 3) , ( 5, 4) , (5, 5) , ( 5 , 6) ,
(6 , 1) , ( 6 , 2) , (6 , 3) , ( 6, 4) , ( 6, 5) , ( 6 , 6) }
Possible Sum are from 2 to 12
Sum Cases Frequency Probability
2 { (1 , 1)} 1 1/36
3 { (1,2),(2,1)} 2 2/36 = 1/18
4 { (1,3),(2,2)(3,1)} 3 3/36 = 1/12
5 { (1,4),(2,3),(3,2)(4,1)} 4 4/36 = 1/9
6 { (1,5),(2,4),(3,3)(4,2)(5,1)} 5 5/36
7 { (1,6),(2,5),(3,4)(4,3) ,(5,2),(6,1)} 6 6/36 = 1/6
8 { (2,6),(3,5),(4,4)(5,3)(6,2)} 5 5/36
9 { (3,6),(4,5),(5,4)(6,3)} 4 4/36 = 1/9
10 { (4,6),(5,5)(6,4)} 3 3/36 = 1/12
11 { (5,6),(6,5)} 2 2/36 = 1/18
12 { (6 ,6)} 1 1/36
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