Putnam 2014 A4

Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X] = 1$, $E[X^2] = 2$, and $E[X^3] = 5$. (Here $E[Y]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X = 0$.