I should know better, but here's another tough bonus stumper. We had a memorable last day of school before the holidays. We went caroling at the Solvang Lutheran Home; we had our school gift exchange; we all went to see the Fellowship of the Ring movie premiere; and we ended at my house with a relaxed staff party. Busy day, but this interesting stumper came up to bother me!
At Dunn Middle School, we play "Secret Santa" for the holidays. We write our names on slips of paper and put them in a hat, about 20 kids per class, and each student draws a secret name. On the last day of school before vacation, we present our small gifts and homemade cards. Someone presents their gift to start, then the recipient, and so on until we're done. This year the 6th and 8th grades went through their classes one by one, but the 7th grade gift exchange broke twice when a student was called who had already gone. What is the probability of getting through a chain of 20 names unbroken? What about n names?
DMS students drawing Secret Santa names and exchanging gifts. If someone draws their own name, they replace it and draw again. (What if the last person draws their own name? It hasn't happened yet!) Each class of about 20 kids exchanged gifts among themselves. Two of our three classes got through the entire gift exchange in one chain of giving and receiving. The third class got through about two-thirds of the kids, but then had to start again after the starting person was called before the end. Then they had to start again for the last two kids who had drawn each other's names. Is this the expected result? What are the probabilities?
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