How this problem came about:
There's 10 of us participating in a Secret Santa gift exchange. The issue is that someone might draw his/her own name and we start over.
My solution:
One person at a time draws a name from the "bowl" and that person needs to make sure that he/she has not drawn himself/herself before the next person's turn. If someone draws his/her own name, half of those who already drew a name should replace their drawn names back to the "bowl". For example, say I am the sixth person to draw a name and I draw my own, then I'll ask 2 of those who already drew a name to put theirs back and take part in the drawing once more. (Round down the half of 5 to get 2.)
Definition:
redraw - each time that a person draws his/her name and others put their names back
What I found out:
Whenever I simulate this process 1,000,000 times, the expected number of redraws is 1.36 or 1.37. See https://onlinegdb.com/ryuUmT-6S.
Can you prove this algebraically?