two dice are thrown simultaneously. let x denote the number of sixes. find the probability distribution of x. also find the mean and variance of x, using probability distribution table.
Answers
If two dice are thrown simultaneously, the sample space can be depicted as follows; S = ⎧⎩⎨⎪⎪⎪⎪⎪⎪1,12,13,14,15,16,11,2..............6,21,36,31,46,41,56,51,66,6⎫⎭⎬⎪⎪⎪⎪⎪⎪{1,11,21,31,41,51,62,1....3,1...4,1....5,1...6,16,26,36,46,56,6}
Let X be the random variable for our problem. X can assume the values of 0, 1 and 2.
P (X = 0) = P (no sixes) = ⎧⎩⎨⎪⎪⎪⎪1,12,13,14,15,11,2..............1,31,41,5⎫⎭⎬⎪⎪⎪⎪{1,11,21,31,41,52,1....3,1...4,1....5,1...} = 25362536
P (X = 1) = P ( 6 on 1st die and no 6 on 2nd OR 6 on 2nd die and no 6 on 1st) = {1,66,12,66,23,66,34,66,45,66,5}=1036{1,62,63,64,65,66,16,26,36,46,5}=1036
P (X = 2) = P (2 sixes) = {6,66,6} = 136136
Given the above distribution, the mean can be calculated as follows:
∑(Xi×P(Xi))=0×2536∑(Xi×P(Xi))=0×2536+1×1036+1×1036 +2×136=1236=13