Exploring changing sentiment with simple probability thinking.

Since signs of President Joe Biden’s decline have become increasingly difficult to ignore, the probability that former President Donald J. Trump will win a second, non-contiguous term has increased substantially. This short note examines how this shift occurred and encourages readers to watch for a potential future shift if Biden drops out of the race under pressure from Democrats.

Prior to the debate, former President Trump enjoyed a small edge (much higher in the battleground states) over President Biden, but we can assume for our purposes that the probability was about 40/60 (Biden/Trump) in an election contest between the two.

After Biden’s disastrous debate performance and unsatisfying attempts to shift the narrative away from his ability to govern, we might see the following simplified two scenarios:

1. Trump will face Biden in November.

2. Trump will face Harris and her running mate in November.

Let’s assume that Trump’s chances against Biden if Biden is on the ballot in November is now 80%, and the probability that Biden is on the ballot in November is 50%. The 80% probability is based on the fact that as of early July, former President Trump has an approximately 4-point lead over President Biden in the key battleground states of Michigan, Ohio, Pennsylvania, and Wisconsin (data from fivethirtyeight.com). Then we have p(Trump wins | Biden on ballot) = 0.80 and p(Biden on ballot) = 0.50, so p(Trump wins & Biden on ballot) = 0.80 x 0.50 = 0.40.

We can further assume that if Trump faces Harris (and her chosen running mate), Trump’s chances are even at 50%, and the probability that Vice President Harris is on the ballot in November is 50%, where we assume for simplicity that this is the only alternative. Then we have p(Trump wins | Harris on ballot) = 0.50 and p(Harris on ballot) = 0.50, so p(Trump wins & Harris on ballot) = 0.50 x 0.50 = 0.25.

Since these outcomes are all of those (given our assumptions) under which former President Trump wins, the total probability that former President Trump wins is p(Trump wins & Biden on ballot) + p(Trump wins & Harris on ballot) = 0.40 + 0.25 = 0.65. Thus, under the current uncertainty, former President Trump would be favored almost 2:1 to win the 2024 Presidential election.

If Biden continues in the race, the probability that Biden drops out will likely fall, increasing the perceived probability that former President Trump faces Biden on election day. Absent an increase in the probability that Biden prevails, we will experience an increase in the perceived probability of a victory for former President Trump from p = 0.65 toward the probability that former President Trump beats President Biden given that President Biden is on the ballot in November, p = 0.80.

If Biden drops out of the race, the probability that former President Trump faces Harris will rise to 1.0 under our assumptions. Absent an increase in the probability that Trump prevails against Harris, we will experience a decrease in the perceived probability of a victory for former President Trump from p = 0.65 toward the probability that former President Trump beats Vice President Harris given that Vice President Harris is on the ballot in November, p = 0.50.

Watch for the change in “election sentiment” that will likely be associated with these two possibilities.

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