In: Statistics and Probability
Two dies are rolled simultaneously, thus, the sample space S is given by
a]
S = {( 11 ),( 12 ),( 13 ),( 14 ),( 15 ),( 16 ),( 21 ),( 22 ),( 23 ),( 24 ),( 25 ),( 26 ),( 31 ),( 32 ),( 33 ),( 34 ),( 35 ),( 36 ),( 41 ),( 42 ),
( 43 ),( 44 ),( 45 ),( 46 ),( 51 ),( 52 ),( 53 ),( 54 ),( 55 ),( 56 ),( 61 ),( 62 ),( 63 ),( 64 ),( 65 ),( 66 )}
#S = number of observations in sample space = 36
where, ( i j ) : i on black die and j on red die
The events are
A : Even number occurred on black die ; in ( i j ) i is even
A = {( 21 ),( 22 ),( 23 ),( 24 ),( 25 ),( 26 ),( 41 ),( 42 ),( 43 ),( 44 ),( 45 ),( 46 ),( 61 ),( 62 ),( 63 ),( 64 ),( 65 ),( 66 )}
#A = 18
B : Odd number occurred on red die ; in ( i j ) j is odd
B = {( 11 ),( 13 ),( 15 ),( 21 ),( 23 ),( 25 ),( 31 ),( 33 ),( 35 ),( 41 ),( 43 ),( 45 )( 51 ),( 53 ),( 55 ),( 61 ),( 63 ),( 65 )}
#B = 18
C : Sum of numbers on both dies is 10 ; ( i j ) is such that i + j = 10
C = {( 46 ),( 55 ),( 64 )}
#C = 3
b]
P(A) = #A / #S = 18/36 = 1/2 = 0.5
P(B) = #B / #S = 18/36 = 1/2 = 0.5
P(C) = #C / #S = 3/36 = 1/12
c]
(A and B) = {( 21 ),( 23 ),( 25 ),( 41 ),( 43 ),( 45 ),( 61 ),( 63 ),( 65 )} ; #(A and B) = 9
P(A and B) = #(A and B) / #S = 9/36 = 1/4 = 0.25
P(A)*P(B) = 0.5 * 0.5 = 0.25
Since, P(A and B) = P(A)*P(B) , Events A and B are INDEPENDENT
d]
(A and C) = {( 46 ),( 64 )} ; #(A and C) = 2
P(A and C) = #(A and C) / #S = 2/36 = 1/18
P(A)*P(C) = (1/2) * (1/12) = 1/24
Since, P(A and C) is not equal to P(A)*P(C) , Events A and C are NOT INDEPENDENT