When an event perceived as statistically impossible occurs, imagination runs rampant.
One such event is the parallel biographical features of Abraham Lincoln and John F. Kennedy, two American presidents separated by a century. These features are often presented as statistically impossible, to the point of being presented as proof that we live in a simulation.
The table below presents a list of true coincidences (the list of false coincidences is much longer)
| Lincoln | Kennedy | |
| Year elected to congress | 1846 | 1946 |
| Year elected to presidency | ‘60 | ‘60 |
| Lost a child as president | Yes | Yes |
| Killed on a | Friday | Friday |
| Assassin escape | Booth ran from a theater | Oswald ran to a theater. |
| Killed in | Ford Theater building | Ford Lincoln car |
| Succeeded by | President Johnson, Born in ‘08 | President Johnson, Born in ‘08 |
Human cognition is highly attuned to pattern recognition, often interpreting coincidences as meaningful even when they result from random variation. This tendency, known as apophenia, contributes to the perception of design or significance where only chance operates.
This document evaluates whether such patterns represent genuine statistical anomalies or whether they are more accurately understood as products of retrospective attention. The goal is not to affirm or dismiss these coincidences categorically, but to assess the methodological basis for treating them as improbable and to examine how probability should be applied to post hoc historical observations.
Naive use of mathematics would easily give rise to a hard question: Are these events so impossibly improbable that they would be only possible if we lived in a simulation? However, as we will see, correct use of math allows to see that the coincidences are improbable, but well within the limits of possibilities.
This is a lecture about the law of total probability for a probability course.
Probability Calculation for Lincoln-Kennedy
Coincidences
To evaluate whether the Lincoln–Kennedy coincidences are statistically surprising, we treat each matched item as an independent event and estimate the probability of each. This is an idealization, but it allows for a consistent reference point. We multiply the probabilities to estimate the overall likelihood of such a pattern appearing by chance.
Let’s call $P_{\text{direct}}$ the direct probability of all coincidences. The events and their estimated probabilities are:
- Elected to Congress exactly 100 years apart: $$P_1 = \frac{1}{20}$$
- Elected to presidency in a year ending in ’60: $$P_2 = \frac{5}{51}$$
- Lost a child while president: $$P_3 = \frac{8}{46}$$
- Both killed on a Friday (assuming uniform distribution of days): $$P_4 = \left(\frac{1}{7}\right)^2$$
- Mirrored theater escape pattern. This is a rough estimate based on the rarity of thematic reversals in public assassination events: $$P_5 = 0.01$$
- “Ford” coincidence (Ford Theater / Lincoln made by Ford). This estimate reflects the general rarity of brand or place-name overlaps: $$P_6 = 0.005$$
- Succeeded by a Johnson born in a year ending in ’08, based on surname frequency (0.81%) and uniform distribution of birth years: $$P_7 = 0.0081 \times \frac{1}{100} = 8.1 \times 10^{-5}$$
Assuming independence, the combined probability is:
$$P_{\text{direct}} = P_1 \times P_2 \times P_3 \times P_4 \times P_5 \times P_6 \times P_7 $$
Substituting values:
$$P_{\text{direct}} = \frac{1}{20} \times \frac{5}{51} \times \frac{8}{46} \times \left(\frac{1}{7}\right)^2 \times 0.01 \times 0.005 \times 8.1 \times 10^{-5} $$
Calculating numerically:
$$ P_{\text{direct}} \approx 0.05 \times 0.098 \times 0.174 \times 0.0204 \times 0.01 \times 0.005 \times 8.1 \times 10^{-5} \approx 6.9 \times 10^{-14} $$
This result corresponds to approximately 1 in 14.5 trillion. However, this is a post hoc computation—many other combinations of historical facts could have appeared equally surprising. When adjusted for the number of patterns we might have searched for, the apparent improbability diminishes considerably. This is not a violation of probability theory, but an illustration of how selection bias affects our intuitions about coincidence.
Adjustment Using the Law of Total Probability
The raw probability of the Lincoln–Kennedy coincidences was computed under the assumption that the specific pattern observed was the only one of interest. However, this assumption ignores the fact that countless other patterns could have also appeared statistically surprising. To account for this, we apply the law of total probability across a large space of possible narrative comparisons.
Let $A$ be the event of observing the specific Lincoln–Kennedy pattern. Let $$H_1, H_2, \ldots, H_n$$ represent a partition of the space of all possible coincidence narratives involving pairs of presidents—such as similarities in names, dates, offices, death circumstances, and so on.
By the law of total probability:
$$ P(A) = \sum_{i=1}^{n} P(A \mid H_i) \cdot P(H_i) $$
This tells us that the true probability of observing a striking pattern like this depends on both the likelihood of a specific coincidence given a narrative structure, and the overall probability that we would consider that structure relevant in the first place.
Suppose there are $$n = 1,000,000$$ plausible narrative structures, and each has a very small individual probability—say, $$P(A \mid H_i) = 10^{-12}.$$ Then the probability that at least one such pattern appears among all these possibilities is:
$$ P(\text{at least one rare pattern}) = 1 – (1 – 10^{-12})^{10^6} $$
This choice of one million possible patterns, each with a probability of $10^{-12}$, is arbitrary but conservative. Choosing fewer patterns or higher probabilities would make the Lincoln–Kennedy coincidences even more likely to occur by chance.
This is the probability of observing at least one rare pattern, assuming:
- There are $n = 10^6$ possible narrative patterns,
- Each has a small probability $p = 10^{-12}$,
- And the patterns are statistically independent.
Using the exponential approximation:
$$ (1 – p)^n \approx e^{-np} $$
we get:
$$ (1 – 10^{-12})^{10^6} \approx e^{-10^{-6}} $$
Therefore,
$$ P(\text{at least one}) = 1 – (1 – p)^n \approx 1 – e^{-10^{-6}} $$
Now we apply the Taylor series expansion for small $x$:
$$ e^{-x} = 1 – x + \frac{x^2}{2} – \frac{x^3}{6} + \cdots $$
Dropping higher-order terms for small $x$, we get:
$$ 1 – e^{-x} \approx x. $$
So, in our case:
$$ 1 – e^{-10^{-6}} \approx 10^{-6} .$$
This approximation lets us estimate the probability that some rare pattern appears out of a million possibilities without computing the exact value. It is accurate for small values of $x$ and avoids unnecessary numerical complexity.
So even though each individual pattern is extremely unlikely, the probability that some striking pattern emerges from scanning a million possibilities is about:
$$ P_{\text{adjusted}} \approx \frac{1}{1,000,000} $$
This result is far more likely than the raw probability of the Lincoln–Kennedy coincidences, which we earlier estimated as:
$$ P_{\text{direct}} \approx 6.9 \times 10^{-14} $$
The law of total probability highlights a key insight: retrospective pattern recognition inflates the perceived improbability of observed coincidences. When the vast number of narrative alternatives is taken into account, such coincidences become far less mysterious.
Comparison: Lincoln–Kennedy Coincidences vs. Powerball Odds
To illustrate just how small the raw probability of the Lincoln–Kennedy coincidences is, we can compare it to a well-known low-probability event: winning the Powerball jackpot.
To win the Powerball, a player must choose 5 correct numbers from 1 to 69 (white balls) and 1 correct number from 1 to 26 (red ball). The total number of possible combinations is:
$$ \text{Combinations} = \binom{69}{5} \times 26 $$
Computing the binomial coefficient:
$$ \binom{69}{5} = \frac{69 \times 68 \times 67 \times 66 \times 65}{5 \times 4 \times 3 \times 2 \times 1} = 11,\!238,\!513 $$
Multiplying by the 26 possibilities for the red Powerball:
$$ \text{Total combinations} = 11,\!238,\!513 \times 26 = 292,\!201,\!338 $$
The probability of winning the Powerball in a single ticket is:
$$ P_{\text{Powerball}} = \frac{1}{292,\!201,\!338} \approx 3.42 \times 10^{-9} $$
In contrast, the combined probability we estimated for the Lincoln–Kennedy coincidences—assuming independence and using conservative estimates—was:
$$ P_{\text{direct}} \approx 6.9 \times 10^{-14} $$
At face value, this would imply that the coincidences are about 50,000 times less likely than winning Powerball. But this is misleading, because the Lincoln–Kennedy probability is a post hoc estimate based on observed patterns, while the Powerball odds are computed a priori based on fixed rules of the game.
Once we apply the law of total probability and account for the fact that we could have noticed any of millions of such historical patterns, the effective adjusted probability for seeing some striking presidential coincidence becomes:
$$ P_{\text{adjusted}} \approx 10^{-6} $$
This is nearly 3,000 times more likely than winning the Powerball lottery.
🤡In other words, if you find the Lincoln–Kennedy similarities surprising, you should find it far more surprising that someone actually wins the lottery.🤡
