What's the probability of a coin toss based on testimonies

Question

Here's a problem:

$1000$ randomly selected people are witnessing coin toss, without any instructions about it, one person at a time. There is no claim whether the coin is rigged or not. You interview each person after their witness and everybody you interview tells you that the coin flipped to heads. You get to witness $1001th$ coin toss yourself, performed with the same coin. What's the probability that it will be heads?

My solution:

Most people don't lie most of the time, at least not in situations like this one, so if anybody lied about their witness lies were fewer than truths. I have a $1000$ statements that say that coin flipped to heads, where I can assign probability that less than $50$% of the statements are not true. That means that at least $501$ statements are true, so I conclude that there's at least somewhat more than $50$% chance that $1001th$ coin toss will be heads, so I say heads.

Is this logic correct? If it is, can there be more precise answer? Is there maybe an almost $100$% chance that $1001th$ coin flip will be heads?

Answer 1

A Bayesian approach may be appropriate:

Prior: 'No claim' might translate into saying that a prior distribution on $\theta = P(\text{Heads})$ is $\theta \sim \mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1).$

Likelihood: If you believe the first 100 witnesses are truthful, the binomial likelihood is proportional to $\theta^{100}.$

Posterior: Then according to Bayes' Theorem for 'conjugate prior and likelihood', we have

$$p_{\theta|X}(\theta | X) = p_\theta(\theta) \times p_{X|\theta}(X|\theta) \propto \theta^0(1-\theta)^0 \times \theta^{100}(1-\theta)^0 \propto \theta^{101 - 1}(1-\theta)^{1-1}.$$

where the right-hand member is proportional to the density pf $\mathsf{Beta}(101,1).$ Thus $P(\theta < 1/2) \approx 4 \times 10^{-31},$ is essentially $0.$ In fact, $P(\theta < .95) = 0.0056.$ Your Bayesian self can feel comfortable betting that the $101$st flip will result in Heads again.

pbeta(.5, 101, 1)
[1] 3.944305e-31
pbeta(.95, 101, 1)
[1] 0.005624503

If you believe all 1000 witnesses, the case that the coin is extremely biased for Heads is even stronger, but the probabilities are awkward to to compute and too small for me to ponder.

Note: (a) The symbol $\propto$ is read 'proportional to'. (b) The subscripts on the density and conditional density functions denoted by $p$'s are unnecessary and often omitted, but some people enjoy seeing them, so I have included them.