Likelihood of malicious nodes in a shard

in his post https://vitalik.ca/general/2021/04/07/sharding.html, V. Buterin said: “Certainly, they can wait until they get lucky and get 51% in a single shard by random chance despite having less than 50% of the total stake, but this gets exponentially harder for attackers that have much less than 51%. If an attacker has less than ~30%, it's virtually impossible.“

Now I tried to calculate the exact likelihood of the event e that in at least one shard more than 50 % the shard’s nodes are malicious depending on:

• the total number of nodes N,
• the number of malicious nodes M,
• the number of shards S.

The nodes are assigned randomly to each shard and each shard contains the same number of nodes.

I discussed this with colleagues, but we couldn’t find a solution, and I hope to find an answer here. Our approach was to calculate the total number of all outcomes / number of e = likelihood. But we got stuck with number of e.

Thank you

• If you translate this into a function, people at the Mathematics stack Exchange could help you. Apr 17 '21 at 1:51