There's no natural solution to this problem. In a sense, this is because you've specified both too much and too little. If you'd specified the probability that there will be more than $n$ goals for all non-negative integers $n$, then it would be straightforward to calculate the probability that there will be exactly $n$ goals for all $n$. On the other hand, if you'd specified only one or two probabilities, one might use a model and use the specified value(s) to determine its parameter(s). For instance, a natural model for the number of goals would lead to the Poisson distribution. This has a single parameter (usually denoted by $\lambda$), which you could fix by specifying a single probability. One might consider other distributions, such as a binomial distribution or a normal distribution truncated at $0$. However, all the obvious candidates have at most $2$ parameters, so they would be overdetermined by the three values you specify. You'd have to make some specific and probably to some degree arbitrary assumptions to come up with a model that has $3$ parameters to fit.
You could consider performing a non-linear least squares fit of a Poisson distribution to the probabilities you've specified. The data fit that model reasonably well; here's a table of values for $\lambda=5.25$ that gets within $0.03$ of your probabilities; this corresponds to an expected number of $5.25$ goals. (Here $\Gamma(n,\lambda)/\Gamma(n)$ is the probability that there will be less than $n$ goals.)