Since my last update, none of the Cook ratings have changed. Here's an updated chart, which differs from the previous one only by statistical noise. For an authoritative probability distribution, see Sam's.

UPDATE July 23, 2016

Since my last update, several Cook ratings have changed (all to the Democrats' favor). Here's an updated chart. For an authoritative probability distribution, see Sam's.

UPDATE July 8, 2016

In the time since my last update, one rating has changed from "lean D" to "likely D." Here's an updated chart. Because it only takes a few seconds, I decided to update the chart even though Sam Wang is publishing his model now. Note, Sam's (infinitely better) model yields a substantially different distribution from the one below. Using my heuristic assumptions, control of the Senate is currently a tossup; but if you look at actual poll data (which Sam does), the Republicans are still favored to retain the majority.

UPDATE June 20, 2016

In the time since my original post below, one rating has changed from "likely D" to "solid D." Here's an updated chart.

I see that Sam Wang is publishing his model now, so I don't need to update this chart again.

*Original post:*

The Cook Political Report rates the competitiveness of Senate races using terms like "solid," "tossup," and so on. Currently,

- Of the 10 Democratic seats in play, 7 are solid D, 1 is likely D, 1 leans D, and 1 is a tossup.
- Of the 24 Republican seats in play, 11 are solid R, 6 are likely R, 1 leans R, and 6 are tossups.

The Democrats have fewer seats in play, and the Democratic seats are generally safer. Obviously this situation favors the Democrats. Is it enough to flip the Senate? Currently, the Republican majority is 54–46 (awarding the two Independent senators to the Democrats, since that's who they caucus with).

For the sake of building a toy model, I assigned probabilities to the Cook Report's verbal ratings as follows:

- For a tossup race, I assigned both outcomes equal probability ½.
- For leaning seats, I assigned probability ⅔ to the outcome matching the lean.
- For likely seats, I assigned probability 0.85 to the likely outcome.
- For solid seats, I assigned probability 1 to the indicated outcome.

These parameters aren't based on any firsthand knowledge of the races in question, they're just ballpark figures.

With probabilities assigned, it is easy to simulate the election many times on a computer. Assuming no coding errors (I did this on my lap during jury duty today), we obtain a probability around 24% of the Senate flipping to a majority for the Democrats.

Also, the probability of a fifty-fifty split is 20%, in which case the Vice President delivers a majority for whichever party wins the Presidential election.

Here is the frequency distribution of outcomes based on a million runs of the election. The blue bars represent flipping scenarios, and you can check that their fractions add up to 24%.

Now, don't take this analysis to the bank or anything. The Cook data are bound to change as time goes on, and on top of that, the probabilities that I assigned to Cook's verbal ratings are somewhat arbitrary. For example, if "likely" means probability 0.75 (instead of 0.85), then the chances of a flip increase from 24% to 34%. Intuitively, more volatility is bad for Republicans because they have more seats in play. Conversely, less volatility favors them; if "likely" is 0.95 and "lean" is 0.9, then the chances of a flip are only 12%.