Phages kill bacteria. One or more phages can attack one bacterium, in which case the bacterium dies, and the phage(s) with it. Thus each phage can attack only one bacterium.
Collectively and over the long run, the phages may have an advantage, because in an attack, a phage injects its DNA into the bacterium, and the bacterium dies in the process of making many more phages. This is how phages procreate. But bacteria that are not attacked eventually divide to increase their population. All of this takes some time, so we do not consider second generation phages or bacteria in this problem.
A beaker contains a large number of phages and the same number of bacteria. If the phages were "efficient," then each of them could kill one bacterium, and the bacteria would all be wiped out. But the (first generation) phages attack bacteria totally at random, until all of the phages have died. The question is, under such random attack, what percent of the bacteria survive attack by these phages.
Recently on this site there have been questions about when to use the binomial distribution to model a practical situation and when to use the Poisson. In that context, the binomial is often the exact one and the Poisson an approximation. I offer this as an example where a Poisson model seems the natural choice.
Let $X \sim POIS(1)$ be the number of phages attacking a randomly chosen bacterium. The mean is $\lambda = 1$ because there are equal numbers of bacteria and phages (so there is, on average, one phage per bacterium).
Then the probability a random bacterium survives is $P\{X = 1\} = 1/e = 0.3679$ (as anticipated in a comment), so that about 37% of the bacteria survive.
Notes: (1) Because the number $n$ of phages (or of bacteria) is unknown, it is not straightforward to use a binomial model without making additional assumptions. (2) In order to model the next generation, we would have to know the distribution of the number of phages a bacterium expels when it dies, and the rate of dividing of remaining bacteria. (3) Overall, it is not wise for phages to be so "efficient" as to wipe out all bacteria because bacteria are the only means of reproduction for any remaining phages. (4) While suffering from a bacterial infection, perhaps it is some small comfort to know bacteria have their own scourges.