Large language models (LLMs) exhibit the ability to generalize given few-shot examples in their input prompt, an emergent capability known as in-context learning (ICL). We investigate whether LLMs use ICL to perform structured reasoning in ways that are consistent with a Bayesian framework or rely on pattern matching. Using a controlled setting of biased coin flips, we find that: (1) LLMs often possess biased priors, causing initial divergence in zero-shot settings, (2) in-context evidence outweighs explicit bias instructions, (3) LLMs broadly follow Bayesian posterior updates, with deviations primarily due to miscalibrated priors rather than flawed updates, and (4) attention magnitude has negligible effect on Bayesian inference. With sufficient demonstrations of biased coin flips via ICL, LLMs update their priors in a Bayesian manner. Code and visualizations are available on the [project page](https://ai-climate.berkeley.edu/llm-coin-flips/).