Using bandit algorithms to design response-adaptive trials can optimize participant outcomes, but poses major challenges for statistical inference. Recent attempts to address these challenges typically impose restrictions on the exploitative nature of the bandit algorithm and require large sample sizes to ensure asymptotic guarantees. However, large experiments generally follow a successful pilot study, which is tightly constrained in its size or duration. In this work, we tackle the problem of hypothesis testing in finite samples. We illustrate an innovative hypothesis testing procedure, uniquely based on the allocation probabilities of the bandit algorithm, and theoretically characterise it when applied to Thompson sampling.