# Yerlan

1. Prove that $\displaystyle P(ABC) = P(B)P(C|B)P(A|BC)$ . Solution.
We know that $\displaystyle P(XY) = P(Y)P(X|Y)$ . Let consider $\displaystyle X=C$ and $\displaystyle Y=B$ then $\displaystyle P(BC) = P(B)P(C|B)$ . Also $\displaystyle X=A and Y=BC$ then $\displaystyle P(ABC) = P(BC)P(A|BC)=P(B)P(C|B)P(A|BC)$ .
2. What is the probability that the sum of two dice is odd with neither being a 4? Solution. Let introduce $\displaystyle N1$ - first dice number, $\displaystyle N2$ - second dice number Let denote event
• $\displaystyle A$ is case when $\displaystyle N1+N2$ is odd
• $\displaystyle B$ is case when $\displaystyle N1 or N2$ is not equal to 4
So $\displaystyle P(A|B) = P(AB)/P(A)=(12/18)/(18/36)=1/3$ $\displaystyle P(AB)=12/18$ and $\displaystyle P(A)=18/36$ it can be seen from table). table