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))
[self-study]
tag & read its wiki. $\endgroup$