If 10 coins were flipped

If 10 coins were flipped, what would be the probability of 9 of them landing on heads? Would it be 1/512 or would the probability be different considering that 1 coin failed at landing on heads? Thank you

• $\frac{1}{512}=\frac{1}{2^9}$ is the probability that in nine flips, all nine are heads, or equivalently that in ten flips, the first nine are heads and the tenth is either. – JMoravitz Sep 6 '16 at 18:26

There are $2^{10}$ (equally probable) outcomes of $10$ coin flips. There are $10$ different possible outcomes where exactly 9 coins land on heads: $HHHHHHHHHT$, $HHHHHHHHTH$, ... $THHHHHHHHH$. So the probability is $\frac{10}{2^{10}}$.
Define a coin ending up being heads a success, with probability $p$. Then, by the binomial distribution, we have
$$\binom{10}{9}p^{9}(1-p)^{1}.$$
If the coin is fair, $p=\frac{1}{2}$ and the probability becomes $$\binom{10}{9}\left(\frac{1}{2}\right)^{10}=10\left(\frac{1}{2}\right)^{10}\approx 1\%.$$