Final answer:
A court would likely require Ann to follow through on the promise under the doctrine of promissory estoppel.
Step-by-step explanation:
In this circumstance, a court would likely require Ann to follow through on the promise under the doctrine of promissory estoppel. Promissory estoppel is a legal doctrine that prevents a person from going back on their promise if the promisee relied on the promise to their detriment. In this case, Mary relied on Ann's promise to pay for the drinks and it would be unfair to allow Ann to change her mind after Mary relied on the promise.