Even number of heads with one fair coin in the mix

You toss $n$ coins at once. Exactly one of them is fair; each of the remaining $n-1$ comes up heads with probability $\lambda$, where $0<\lambda<1$. What is the probability that the total number of heads is even?

$1/2$. Condition on the other $n-1$ coins: whatever the parity of their heads count, the fair coin flips the overall parity with probability exactly $1/2$. So the total is even with probability $\tfrac12$ regardless of $\lambda$ and $n$.