A throwback to Part I and why live draws can absolutely be rigged.
When I heard the news that the first day of the Democratic debates on July 30 featured only white candidates and that all of the non-white candidates were scheduled for the second day, I knew something was off (I wasn’t the only one who had suspicions). Admittedly, I hadn’t been following the debates very closely, but my gut told me that even in such a large primary field, there were enough minority candidates that the likelihood of such an outcome happening by pure chance was quite slim.
I decided to get to the bottom of things, despite combinatorics being one of my weakest areas in math growing up. To start, I had to confirm the number of white vs. non-white candidates in the debate field. I quickly found out that there were only 5 non-white candidates: Kamala Harris, (black), Cory Booker (black), Julián Castro (Latino), Andrew Yang (Asian), and Tulsi Gabbard (Pacific Islander).
A First Pass
If CNN randomly selected each candidate and their debate day, then we can calculate the total number of ways that 20 candidates can be divided into two groups. Assuming that order matters (i.e. having only white candidates on the first day of debates is different from having only white candidates on the second day), then there are a total of \binom{20}{10}=184,756 possible combinations. Out of those, there are \binom{15}{10} \times \binom{5}{0}=3,003 ways to choose only white candidates on the first day. Therefore, the probability of featuring only white candidates on the first day is \frac{3,003}{184,756}=1.63\%. Not very likely, eh?
CNN, What Were You Thinking?
Interestingly enough, CNN did NOT use a purely random selection process, instead electing to use a somewhat convoluted three-part draw “to ensure support for the candidates [was] evenly spread across both nights.” The 20 candidates were first ordered based on their rankings in the latest public polling, and then divided into three groups: Top 4 (Biden, Harris, Sanders, Warren), Middle 6 (Booker, Buttigieg, Castro, Klobuchar, O’Rourke, Yang), and Bottom 10 (Bennet, Bullock, de Blasio, Delaney, Gabbard, Gillibrand, Hickenlooper, Inslee, Ryan, Williamson).
“During each draw, cards with a candidate’s name [were] placed into a dedicated box, while a second box [held] cards printed with the date of each night. For each draw, the anchor [retrieved] a name card from the first box and then [matched] it with a date card from the second box.”
CNN
In other words, CNN performed a random selection within each of the three groups, and the three draws were independent events.
A New Methodology
To calculate our desired probability under the actual CNN methodology, we need to figure out the likeliness of having only white candidates on the first day for each of the three groups. We can then multiply these probabilities together since the events are independent. For the Top 4 (where Harris is the only non-white candidate), there are \binom{4}{2}=6 total combinations, and \binom{3}{2} \times \binom{1}{0}=3 ways to choose only white candidates on the first day. Therefore, the probability of featuring only white candidates on the first day is \frac{3}{6}=50\%.
For the Bottom 10 (where Gabbard is the only non-white candidate), there are \binom{10}{5}=252 total combinations, and \binom{9}{5} \times \binom{1}{0}=126 ways to choose only white candidates on the first day. Therefore, our desired probability is \frac{126}{252}=50\%.
It should make sense that the probability is 50% for both the Top 4 and Bottom 10, precisely because there is exactly one candidate of color in each group. Think about it for a second: in both scenarios, the non-white candidate either ends up debating on the first day or the second day, hence 50%.
The Middle 6 is where it gets interesting. There are exactly 3 white candidates and 3 non-white candidates. This yields \binom{6}{3}=20 total combinations, but only \binom{3}{3} \times \binom{3}{0}=1 way to choose only white candidates on the first day, or a probability of just \frac{1}{20}=5\%.
Since the three draws are independent events, we can simply multiply the probabilities to get to our desired answer: 50\% \times 50\% \times 5\% = 1.25\%. Even lower than the 1.63% from our first calculation!
One More Twist
Even a casual observer may have noticed that although the first day of debates featured an all-white field, Democratic front-runner Joe Biden was drawn on the second day. This conveniently set up what many media outlets touted as a “rematch” with Senator Kamala Harris, with CNN going so far as comparing the match-up to the “Thrilla in Manila” (I wish I were joking).
The probability of have only white candidates on the first day AND Joe Biden on the second day is 16.67\% \times 50\% \times 5\% = 0.42\%. The only difference between this scenario and the previous one is that within the Top 4, there is only one way to draw both Biden and Harris on the second day out of a total of six possible combinations: \frac{1}{6}=16.67\%.
Validating with Monte Carlo
I wasn’t 100% certain about my mathematical calculations at this point, so I decided to verify them using Monte Carlo simulations. Plus, this wouldn’t be a “Fun with Excel” post if we didn’t let Excel do some of the heavy lifting 🙂
I set up a series of random number generators to simulate CNN’s drawing procedure, keeping track of whether Scenario 1 (only white candidates on the first day) or Scenario 2 (only white candidates on the first day AND Joe Biden on the second day) was fulfilled in each case. Excel’s row limit only let me run 45,000 draws simultaneously, which I then repeated 100 times and graphed as box and whisker plots below:
The simulations yielded an average of 1.26% for Scenario 1 and 0.42% for Scenario 2, thus corroborating the previously calculated theoretical probabilities of 1.25% and 0.42%.
Concluding Thoughts
Numbers don’t lie, and they lead me to conclude that the CNN Democratic Debate Draw was not truly random. The million dollar question, of course, is why? What does CNN gain from having only white candidates on the first day and Joe Biden on the second day (along with all the minority candidates)? As I don’t intend for my blog to be an outlet for my personal political views, I’ll leave out any “conspiracy” theories and leave them as an exercise for you, the reader.
As always, you can find my work here.
This is Post #20 of the “Fun with Excel” series. For more content like this, please click here.
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