Probability 7

Conditions to be considered (only consider the ending digits)
(1) all integers must be odd (ending digits 1,3,5,7,9: 5 values)
(2) all integers must contain at least one integer ending with digit 5

Number of cases that all integers are odd are 5^n
Number of cases that all integers are odd but without an integer ending with digit 5 are 4^n (ending digits 1,3,7,9 only)

Number of cases that all integers are odd and with at least one ending with 5 is 5^n - 4^n

The required probability is (5^n - 4^n)/10^n

[verify with n=2, the probability is 9/100. The cases for the ending digits are 1*5 3*5 5*5 7*5 9*5 5*1 5*3 5*7 5*9 total 9 cases out of 100]

We know that 25>24
5.5>6.4
5/6>4/5
also 4/5>(4/5)^n
So 5/6>(4/5)^n
and 5.5^n - 6.4^n >0

Now consider the difference
(5^n - 4^n)/10^n - [5^(n+1) - 4^(n+1)]/10^(n+1)
=[10.5^n-10.4^n - 5^(n+1) + 4^(n+1)]/10^(n+1)
=(5.5^n - 6.4^n)/10^(n+1) > 0

So the expression (5^n - 4^n)/10^n decreases when n increases.