Roll two fair dice Let A be the event that at least one of t

Roll two fair dice. Let A be the event that at least one of the rolls is a 4 and B be the event that both of the die are the same value. Are A and B independent? Find P (A | B ) and P(B | A)

Solution

The experiment is rolling two fair dice , So the sample sapace i.e. all possible out come can be represented by \'S\'.

S = {11,12,13,14,15,16,21,22,....., 66}

The Modulus of S = |S| = 36

\'A\' represents at least one of the role is 4 then A = {14,24,34,44, 54, 64, 41, 42, 43, 45,46}

|A| = 11, P(A) = 11/36

\'B\' represents  both of the die are the same value i.e. B = {11,22,33,44,55,66} , so |B| = 6 P(B) = 6/36

Answer :1

A and B are not independent , because had the information about any one of the event is known affects the probabilty of occurence of the other event.

Mathematically, AB = {44}, P(AB) = 1/ 36.and P(AB) P(A) *P(B), Therefore they are dedependent.

Answer :2

P(A|B) = P(AB)/P(B) = (1/36)/(6/36) = 1/6, There is a change in the probability of occurrence of the event A to 1/6 from 1/11, if the occurrence of event B is known.

Similarly P(B/A) = P(AB)/ P(A) = (1/36)/ (11/36) = 1/11, There is a change in the probability of occurrence of the event B to 1/11 from 1/6,if the occurrence of event A is known.