Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1,E\left[X^2\right]=2$,and $E\left[X^{3}\right]=5.$(Here $E\left[y\right]$denotes the expectation of the random variable $Y.)$ Determine the smallest possible value of the probability of the event $X=0.$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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