An internal risk analysis after the first of two Boeing 737 MAX airliner crashes showed the likelihood was high of a similar cockpit emergency within months, according to a Federal Aviation Administration official familiar with the details and others briefed on the matter.

The regulator’s analysis, not previously reported, showed that it “didn’t take that much” for a malfunction like the one confronted by the pilots of the Lion Air flight that crashed into the Java Sea last year to occur, one of the people briefed on the analysis...