Here’s a cute one for your intro probability class.

Karen Langley from the Wall Street Journal asks:

What is the probability of the Dow Jones Industrial Average closing unchanged from the day before, as it did yesterday?

To answer this question we need to know two things:

1. How much does the Dow Jones average typically vary from day to day?

2. How precisely is the average recorded?

I have no idea about item #1, so I asked Langley and she said that it might go up or down by 50 in a given day. So let’s say that’s the range, from -50 to +50, then the probability is approximately 1/100 of there being no change, rounded to the nearest integer.

For item #2, I googled and found this news article that implies that the Dow Jones average is rounded to 2 decimal places (e.g., 27,691.49).

So then the probability of the Dow being unchanged to 2 decimal places is approximately 1/10000.

That’s a quick calculation. To do better, we’d want a better distribution of the day-to-day change. We could compute some quantiles and then fit a normal density—or even just compute the proportion of day-to-day changes that are in the range [-10, 10] and divide that by 1000. It should be pretty easy to get this number.

Yet another complexity is that there’s a small number of stocks in the Dow Jones average, so it might be that not all prices are even possible. I don’t think that’s an issue, as it seems that each individual stock has a price down to the nearest cent, but maybe there’s something I’m missing here.

Another way to attack the problem is purely empirically. According to this link, the Dow being unchanged is a “once-in-a-decade event.” A year has approximately 250 business days, hence if it’s truly once in a decade, that’s a probability of 1/2500. In that case, my 1/10000 estimate is way too low. On the other hand, given that prices have been rising, the probability of an exact tie should be declining. So even if the probability was 1/2500 ten years ago, it could be lower now. Also, an approximate ten-year gap does not give a very precise estimate of the probability. All these numbers are in the same order of magnitude.

Anyway, this is a good example to demonstrate the empirical calculations of probabilities, similar to some of the examples in chapter 1 of BDA but with more detail. I prefer estimating the probability of a tied election, or maybe some sports example, but if you like stocks, you can take this one.