Probability that 4 single digit numbers whose sum is 28 contain at least two prime numbers

Question

What is the probability that a list of 4, not necessarily distinct, single-digit numbers, whose sum is 28, will contain at least two prime numbers?

There are not a whole lot of possible "desirable" cases ,but I was trying to see if there's a way to limit my search.

Answer 1

There are only six cases that fulfill the requirements:

{{9, 9, 7, 3}, {9, 9, 5, 5}, {9, 7, 7, 5}, {8, 8, 7, 5}, {8, 7, 7, 6}, {7, 7, 7, 7}}

The total number of cases is 15, so the probability is $0.4$.

Start with the highest value in a set of four is $9$. If the next value is also $9$, then the following two values must be primes, giving $7$ and $3$ or $5$ and $5$.

It is simple to see that $9, 8$ has no two prime completions, to move to $9, 7$. The only completion is then $7, 5$. And we're done with $9$ as the largest digit.

Consider $8$ as the largest digit. The largest second digit is then $8$, which gives a completion of $7, 5$.

And so on....