The Venn diagram shows the probabilities associated with three events XXX, Y Y\,Y and ZZZ, representing students choosing to study different languages. The probability of students studying only X X\,X is 0.15, only Y Y\,Y is 0.2, and only Z Z\,Z is 0.4. The intersection X∩Y X \cap Y\,X∩Y is q q\,q and Y∩Z Y \cap Z\,Y∩Z is ppp. Events X X\,X and Z Z\,Z are disjoint. The probability of studying none of these languages is rrr.
Write down the two events that are mutually exclusive.
Given that P(Y∣Z)=0.2P(Y | Z) = 0.2P(Y∣Z)=0.2, find the value of ppp.
Given also that X X\,X and Y Y\,Y are independent, find the values of q q\,q and rrr.
