A Coin Flip Problem

Question

Suppose I have $n$ fair coins, and I mark one of them for identification. Next I flip the $n$ coins without looking. My friend, who is looking on, now informs me that there were at least $k$ heads flipped.

What is the probability that my marked coin was heads?

Answer 1

Let your coin be $X_1$ and denote sum of heads as $S$.

As I have written in the comment the answers seems to be

$$P(X_1 = 1| S \ge k) = \frac{\sum_{i = k}^{n} \binom{n-1}{i-1}}{\sum_{i=k}^{n}\binom{n}{i}}$$

Here is a plot of theoretical vs sample probabilities with $n = 20$ and 1e^7 trials

We can see that with low values of $k$ we get almost no additional information, thus the probability is close to unconditional $0.5$

Partially recreated code as requested by @Maximilian

library(tidyverse)

coin_flips <- function(n, k) {
    # Create n x k matrix of binary outcomes
    flips <- matrix(as.numeric(rbinom(n * k, 1, 0.5)), ncol = k)
    firsts <- flips[, 1]
    flips <- t(apply(flips, 1, sort, decreasing = T)) # i-th column is an indicator value [S >= i]
                                                      # where S is the sum of heads
    flips <- as.tibble(flips)
    f <- function(x) {
        if (sum(x) > 0) {
            return(sum(x * firsts) / sum(x))
        }
        return(1)
    }
    summary <- flips %>%
        summarise_all(.funs = f)
    colnames(summary) <- 1:k
    return(summary)
}
# Example usage
cf <- coin_flips(1000000, 20)
cf %>% gather %>% ggplot(aes(as.numeric(key), value)) + geom_point() + ylim(c(0.48, 1))