During a recent appearance on “The Daily Show,” the host Trevor Noah asked Christine Lagarde if she was familiar with the “glass cliff.” It is the phenomenon, he said, of handing a job to a woman at a hazardous moment, making her success highly improbable, and giving men some deniability.

Hadn’t Ms. Lagarde landed on a glass cliff when she was appointed head of the International Monetary Fund in 2011, Mr. Noah asked, a time when many countries were still reeling from the financial crisis?

“Correct! You’re right,” Ms. Lagarde replied. The job, she went on, “was intimidating.” But the circumstance — a woman elevated just in time to oversee a fiasco — felt familiar.

“Whenever the situation is really, really bad,” she said, “you call in the woman.”

The woman has been called in again. On Tuesday, Ms. Lagarde was nominated to be the new president of the European Central Bank, the first woman to be chosen for the post. The role will make her one of the most powerful figures in international finance, giving her a leading hand in guiding the world’s second-largest economy. She will succeed Mario Draghi, an Italian economist who will be remembered for steering the eurozone — the 19 countries that use the euro — through a perilous era, including the near bankruptcies of Greece and Spain.