When you flip a coin, you have a 1 in 2 chance of getting heads. The odds stay 1 in 2 for getting heads no matter how many times you flip the coin. The odds of it not coming up tails at least once every 10 times is low, but the largest factor in that is the 1 in 2 chance for it to come up tails, not the 10 flips.
If your chance of contracting an STI is 1 in 1000, it stays a 1 in 1000 chance with every encounter with every partner. That means with each encounter and with every partner, the odds against contracting an STI is 999 to 1, whether you have 10 partners or 1000. Since the actual odds of contracting a fluids transmissible STI when properly using barriers/safer sex practices is actually much lower than 1000-1, the coin flip analogy is even farther from being useful.
I get that it doesn't appear to be logical (I did say it was counterintuitive), but odds are not actually a cumulative thing. If they were, people would win at casinos a helluva lot more often than they actually do. But they don't, because the odds remain the same with each play on the slot machine, no matter how many times you play.