Conditional Probability and the Inverse Fallacy

Why information changes a probability

An unconditional probability is what you would say if you knew nothing else. Pick a random MLB team in March: 12 of 30 make the playoffs, so P(playoffs) is 40%. That number is the answer when you have no information beyond “it’s a major league team in this era.”

A conditional probability is what you would say if you knew one specific additional thing. Once you learn the team is 10-21 through 31 games, you are no longer reasoning about a random team — you are reasoning about a team in a specific category, the “sub-.400 starts” category. The right reference class is no longer all 30 teams; it is the historical set of teams that have started this poorly. Inside that smaller, harsher reference class, the playoff rate is much lower — around 11%.

The vertical bar in the notation P(A | B) is doing all the work. It says: restrict your attention to the cases where B is true, and ask what fraction of those have A. Different B, different reference class, different fraction. Same A.

The same shape of reasoning shows up everywhere. In medicine, P(has the disease | tested positive) and P(tests positive | has the disease) are routinely confused, even by clinicians. The first is what the patient cares about; the second is what the test reports. The first depends on the base rate. In law, the “prosecutor’s fallacy” is the courtroom version of the same swap. In each case, the cure is the same: write the bar correctly, identify which side of it is the condition, and check which question you actually want answered.

Bayes’ theorem, applied to 10-21

The two conditional probabilities — P(A | B) and P(B | A) — are related by Bayes’ theorem. The full statement:

For our case, let A = “makes the playoffs” and B = “starts the season at or below .400 through 31 games.” Then:

To turn this into numbers, we need three pieces of historical data. Working from the last ten seasons (2016–2025, n ≈ 300 team-seasons under varying playoff formats):

Plugging in:

The same answer arrives if we count directly: roughly 4 out of 35 sub-.400 starts made the playoffs, which is about 11%. Bayes’ theorem is a sanity check on the direct count, not a replacement for it. But it tells you something the direct count doesn’t: it shows you, in algebra, exactly how the inverse conditional (P(bad start | playoffs) = 3.3%) is related to the conditional you actually want (P(playoffs | bad start) = 11.3%). They differ by a factor of P(playoffs) / P(bad start) — in this case, about 3.4×. The base rates do that work.

This is the trap that single-case reasoning lays. One memorable comeback — the Nationals, the Astros — can be read as evidence that comebacks are common, when in fact each one is a draw from a low-probability tail. The math does not say the comeback was impossible. The math says the comeback was, before it happened, a roughly one-in-ten event — and that calling it “probable” in advance would have been wrong.

Computing P(playoffs | start) from a CSV

Below is a self-contained Python script that takes a CSV of historical seasons (one row per team-season, with columns for the early-season record and a binary playoffs indicator) and computes both conditional probabilities — the one we want and its inverse — alongside the underlying base rates. It is short on purpose. The whole point of conditional probability is that the arithmetic is trivial; the discipline is in asking the question correctly.

# conditional-probability.py — compute P(playoffs | start) and its inverse
import csv

def load_seasons(path):
    # expects columns: team, year, wins_thru_31, losses_thru_31, made_playoffs (0/1)
    with open(path) as f:
        return list(csv.DictReader(f))

def conditional_probs(rows, threshold=0.400):
    n_total = len(rows)
    n_playoffs = sum(1 for r in rows if int(r["made_playoffs"]))
    n_badstart = sum(1 for r in rows
                     if _winpct(r) <= threshold)
    n_both = sum(1 for r in rows
                 if _winpct(r) <= threshold
                 and int(r["made_playoffs"]))

    P_A   = n_playoffs / n_total                            # P(playoffs)
    P_B   = n_badstart / n_total                            # P(bad start)
    P_AB  = n_both     / n_badstart  if n_badstart else 0  # P(playoffs | bad)
    P_BA  = n_both     / n_playoffs  if n_playoffs else 0  # P(bad | playoffs)

    return {"P(playoffs)": P_A, "P(bad)": P_B,
            "P(playoffs|bad)": P_AB, "P(bad|playoffs)": P_BA,
            "n_total": n_total, "n_badstart": n_badstart,
            "n_both": n_both}

def _winpct(r):
    w, l = int(r["wins_thru_31"]), int(r["losses_thru_31"])
    return w / (w + l) if (w + l) else 0

if __name__ == "__main__":
    rows = load_seasons("seasons_2016_2025.csv")
    out  = conditional_probs(rows, threshold=0.400)
    for k, v in out.items():
        print(f"{k:24s} {v}")

The two conditional probabilities are computed from the same three counts — n_total, n_badstart, and n_both. The numerator is the same in both directions. Only the denominator changes. Switching from P(A | B) to P(B | A) is a one-line change: divide by n_playoffs instead of n_badstart. That single change can move the answer by a factor of three or more, which is why the inverse fallacy is so dangerous and so hard to spot.