Individuals::Individual ID Number should be an auto-entered serial number. Event Information::Individual ID number should not be an auto-entered serial number field.
If you are modifying a record in Event Information, you can use a value list that lists both ID numbers and Names from the Individuals table and use a button for adding a new record in the individual table. This button can both create the new record in individuals and then create a matching record in Event Information by giving te ID number field the same value as that of the newly created Individuals record.
You can find examples of that in this demo file which, coincidentally, models matching up contacts to events in a many to many relationship: https://www.dropbox.com/s/oyir7cs0yxmbn6i/ManyToManywDemoWExtras.fp7
If you are using FileMaker 12, you can open this file from the File menu to get a copy converted to the .fmp12 file format.