Comment by donnerdave

8 hours ago

No one commenting on the inaccuracy (or at least imprecision) of the Python output cited in the article??

"A:T, B:T - chances - H 6.0% | T 94.0% | occurs 34.0% of the time"

By the simplest of math for unrelated events, the chance of both A & B lying about the coin is 20% of 20%, or .2 * .2 = 0.04, or 4.0% ...

The "Let's prove it" section contains the correct analysis, including that our chance of being correct is 80% with two friends.

The code output for three players is similarly flawed, and the analysis slight misstates our chance of being correct as 90.0% (correctly: 89.6%).

Or am I missing something about the intent or output of the Python simulation?

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donnerdave

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eieio  3 hours ago

This is an interesting question!

But no, the python output is correct (although I do round the values). It's counterintuitive but these are two different questions:

    1. What are the odds that both players lie? (4%)
    2. Given that both players say tails, what are the odds that the coin is heads (~6%)

Trivially, the answer for question (1) is 0.2 * 0.2 = 4%

The answer for question (2) is 0.02 / 0.34 = 6%

One way of expressing this is Bayes Rule: we want P(both say tails | coin is heads):

    * we can compute this as (P(coin is heads | both say tails) * P(coin is heads)) / P(both say tails)
    * P(coin is heads | both say tails) = 0.04 (both must lie)
    * P(coin is heads) = 0.5
    * P(both say tails) = 0.04 * 0.5 + 0.64 * 0.5 = 0.34

This gives us (0.04 * 0.5) / 0.34 = 0.02 / 0.34 ~= 6%

I think that might not be convincing to you, so we can also just look at the results for a hypothetical simulation with 2000 flips:

    * of those 2000 flips, 1000 are tails
    * 640 times both players tell the truth
    * 40 times both players lie
    * 680 times (640 + 40) both players *agree*
    * 320 times the players disagree

We're talking about "the number of times they lie divided by the number of times that they agree"

40 / 680 ~= 6%

We go from 4% to 6% because the denominator changes. For the "how often do they both lie" case, our denominator is "all of our coin flips." For the "given that they both said tails, what are the odds that the coin is heads" case, our denominator is "all of the cases where they agreed" - a substantially smaller denominator!

The three players example is just me rounding 89.6% to 90% to make the output shorter (all examples are rounded to two digits, otherwise I found that the output was too large to fit on many screens without horizontal scrolling).