probability of number ending in 1

[perusal]

Question: what is the probability of a randomly selected positive integer raised to an arbitrary positive integer ending in the number 1?

The answer is 0.04 I am unsure how this can be arrived at.

My understanding is that any integer ending in 1 will always end in 1 when raised to any positive integer.
No even number will ever end in 1 when raised to any positive integer.
Numbers ending in 9 will end in a 1 when raised to any multiple of 2
Numbers 3 and 7 will end in 1 when raised to any multiple of 4.
Numbers ending in 5 will never end in a 1.

 

[pka]

Question: what is the probability of a randomly selected positive integer raised to an arbitrary positive integer ending in the number 1?
The answer is 0.04 I am unsure how this can be arrived at.
My understanding is that any integer ending in 1 will always end in 1 when raised to any positive integer.
No even number will ever end in 1 when raised to any positive integer.
Numbers ending in 9 will end in a 1 when raised to any multiple of 2
Numbers 3 and 7 will end in 1 when raised to any multiple of 4.
Numbers ending in 5 will never end in a 1.

What is probability that a randomly selected positive integer ends in $\displaystyle 3\text{ or } 7~?$ Is it $\displaystyle \dfrac{2}{10}~?$
What is probability that a randomly selected positive integral power is a multiple of $\displaystyle 4~?$
Repeat that process for $\displaystyle 1~\&~9$. What do you get?

I see that the provided answer is $\displaystyle 0.04$ and I get $\displaystyle 0.23$ see here.