Question

Find the probability N?
No 4 on first die
No 4 on any of the die
No 3 on the 1 die and 5 on the 2 die
3 and 5 on any of the die
Difference between no is 2

Answer 1

Sample space for two dice

$$\def\arraystretch{1.5} \begin{array}{c} (1,1) & (1,2) & (1,3) & (1,4) & (1, 5) & (1,6) \\ \hline (2,1) & (2,2) & (2,3) & (2,4) & (2, 5) & (2,6)\\ \hline (3,1) & (3,2) & (3,3) & (3,4) & (3, 5) & (3,6) \\ \hline (4,1) & (4,2) & (4,3) & (4,4) & (4, 5) & (4,6) \\ \hline (5,1) & (5,2) & (5,3) & (5,4) & (5, 5) & (5,6) \\ \hline (6,1) & (6,2) & (6,3) & (6,4) & (6, 5) & (6,6) \end{array}$$

No 4 on first die :

$(4,1), (4,2),(4,3),(4,4), (4, 5),(4,6)$

$$P(4 \ on\ first)={6 \over 36}={1 \over 6}$$

No 4 on any of the die:

$(4,1), (4,2),(4,3),(4,4), (4, 5),(4,6),(1,4), (2,4),(3,4),(5,4), (6, 4)$

$$P(4\ on \ any)={11 \over 36}$$

No 3 on the 1 die and 5 on the 2 die:

$(3,5)$  

$$P(3\ on \ first\ and \ 5 \ on\ second)={1 \over 36}$$

3 and 5 on any of the die:

$(3,5), (5,3)$

$$P(3\ and\ 5\ on \ any)={2 \over 36}={1\over 18}$$

Difference between No is 2:

$(1,3), (2,4),(3,5),(4,6),(3,1),(4,2),(5,3),(6,4)$

$$P(difference\ is\ 2)={8 \over 36}={2\over 9}$$