Daily Bulletin

The Conversation

  • Written by Rheanna Mainzer, Melbourne Early Career Academic Fellow (Statistics Tutor), University of Melbourne

Last Christmas, my family gathered to organise our Kris Kringle. My sister drew her husband, but they were already buying presents for each other, so we decided to draw again. No one in my family (except me) is particularly interested in mathematics or statistics, but my brother fatefully asked:

What are the chances that all the partners draw each other?

At the time, my family consisted of eight people: my mother and her partner, my older sister and her husband, my younger brother and his new girlfriend, my grandpa (widowed and never remarried), and me (I was single). So there were three sets of partners and two singles.

I started thinking out loud about how to answer my brother’s question. One way to find a probability is to calculate the fraction

the statistics of Secret Santa The “event of interest” here is that all three partners draw each other’s names and, because there are no other possibilities, my grandpa and I draw each other. This can only happen in one way, so the top line of the fraction above must be 1. the statistics of Secret Santa My grandpa unwrapping his new weed sprayer on Christmas day. Author provided What’s the bottom line? Unfortunately, the bottom line is much trickier to calculate. You want to count the total number of ways that eight people can draw names from a hat, without drawing themselves. For example, one possible event outcome is: I draw my mum, who draws my brother, who draws my sister, who draws her husband, who draws my brother’s new girlfriend, who draws mum’s partner, who draws my grandpa, who draws me. My brother’s new girlfriend interrupted my thinking by asking whether the answer isn’t simply eight factorial (which is written in mathematical notation as “8!”). For those who aren’t familiar, the statistics of Secret Santa My family let out a collective “ooooooh!” at what they perceived as a challenge to my mathematical prowess from my brother’s new girlfriend. Her guess was sensible but not quite right, because it included the outcomes in which someone draws their own name. Without pen or paper handy I decided to file the problem away in the back of my mind and rejoin the conversation, which had swiftly moved on. When in doubt, bring in the experts I found the perfect place to resurrect this problem, at a recent MathsCraft event, which brought mathematicians, teachers and students together to explore problems like a research mathematician. My University of Melbourne colleagues TriThang Tran, Sam Povall and Rhys Bowden are all experts in different mathematics research fields. I couldn’t think of anyone more qualified to count things than these three – surely they must be able to answer my Christmas conundrum. However, a short brainstorm session revealed that the Kris Kringle question is harder than initially expected. But lo, a possible saviour was delivered unto us in the form of Professor Nigel Bean, an expert in applied probability. Like the star guiding the Three Wise Men, hopefully Nigel would be able to guide us puzzled mathematicians to the solution. “That’s easy!” Nigel confidently announced. But seconds later, his face fell as he too came to the realisation that this problem is deceptively difficult. We threw ideas around the table, our voices rising in excitement. This captured the attention of two secondary school maths teachers, Amy Xue and Callum Johnson, who came over to investigate. While Sam brought Amy and Callum up to speed, Nigel and Rhys worked on the whiteboard, drawing tables and defining notation to help them. TriThang took a different approach, working at the table and using pictures to illustrate different outcomes. the statistics of Secret Santa Mathematicians working on the Kris Kringle problem together. From left to right: Nigel, Rhys, Sam, Amy, Callum and TriThang. The solution It took an hour and a half – and a coffee break – but Nigel, Rhys and (independently) TriThang finally arrived at the same solution. The chance that each of the couples in my family draw each other’s names in our annual Secret Santa is 1 in 14,833, or about 0.007%. However, it was Callum who came up with the answer the fastest. Recalling the topic of derangements that he learned nine years ago at university and encountered again in the final question of the 2016 year 12 NSW Extension 2 maths paper, he produced his solution: the statistics of Secret Santa where the bottom line of this equation is just a particularly unusual way of writing 14,833 (the e refers to Euler’s number). I was proud to finally reveal the solution to my brother: it was incredibly unlikely that all three partners would have drawn each other that night at dinner - about as unlikely as tossing a coin 14 times and coming up with 14 heads. However, he wasn’t quite as excited about knowing the answer as I was. (I told you my family aren’t particularly interested in maths.) A Christmas twist Here’s the thing, though: the Kris Kringle question is a known problem, related to the old hats problem. Why didn’t we just Google it? Well, we all really enjoyed working on the problem. And Christmas is supposed to be fun. So I have to agree with Nigel when he asked: “Where is the fun in that?” the statistics of Secret Santa Nigel (left) and Rhys solving the Kris Kringle problem on the whiteboard. Thanks to all the mathematicians who worked on this problem, ACEMS for hosting the event that brought us together, and my family for coming up with the question.

Authors: Rheanna Mainzer, Melbourne Early Career Academic Fellow (Statistics Tutor), University of Melbourne

Read more http://theconversation.com/merry-christmaths-the-statistics-of-secret-santa-127730

Writers Wanted

Dobell Biennial showcases drawing today as we consider its future in the real world


Baby in Doha incident alive but unidentified


'I still cannot get over it': 75 years after Japan atomic bombs, a nuclear weapons ban treaty is finally realised


The Conversation


Prime Minister Interview with Kieran Gilbert, Sky News

KIERAN GILBERT: Kieran Gilbert here with you and the Prime Minister joins me. Prime Minister, thanks so much for your time.  PRIME MINISTER: G'day Kieran.  GILBERT: An assumption a vaccine is ...

Daily Bulletin - avatar Daily Bulletin

Did BLM Really Change the US Police Work?

The Black Lives Matter (BLM) movement has proven that the power of the state rests in the hands of the people it governs. Following the death of 46-year-old black American George Floyd in a case of ...

a Guest Writer - avatar a Guest Writer

Scott Morrison: the right man at the right time

Australia is not at war with another nation or ideology in August 2020 but the nation is in conflict. There are serious threats from China and there are many challenges flowing from the pandemic tha...

Greg Rogers - avatar Greg Rogers

Business News

Guide to Shipping Container Hire

If you are thinking of hiring a shipping container rather than purchasing one, there are many great reasons to do so. It is a more affordable option and when you are done using it for what you neede...

News Co - avatar News Co

Top 5 US Logistics Companies

Nothing is more annoying than having to deal with unreliable shipping companies for your fragile and important packages. Other than providing the best customer service, a logistics company also ne...

News Co - avatar News Co

Luke Lazarus Helps Turns Startups into Global Stalwarts

There are many positive aspects to globalization. It is no secret that those who have been impacted by globalization tend to enjoy a higher standard of living in general. One factor that has led to ...

Emma Davidson - avatar Emma Davidson

News Co Media Group

Content & Technology Connecting Global Audiences

More Information - Less Opinion