2024/018) There are 1000 coins -- 999 are fair, and 1 has heads on both sides. You randomly choose a coin and flip it 10 times. Miraculously, all 10 flips turn up heads. What is the probability that you chose the unfair coin

Probability that you choose one unfair coin = $\frac{1}{1000}$

iI you choose one unfair coin the probability that all 10 heads come = 1

So   Probability that you choose one unfair coin and all heads come = $\frac{1}{1000}$

Probability that you choose one fair coin = $\frac{999}{1000}$

If you choose one fair  coin the probability that all 10 heads come = $\frac{1}{2^{20}} = \frac{1}{1024}$

So   Probability that you choose one fair coin and all heads come = $\frac{999}{1024*1000}$

 So probability that all heads come = $\frac{999}{1024*1000} + \frac{1}{1000} = \frac{2023}{1024000}$

 So probability that coin is unfiar = $\frac{\frac{1}{1000}}{\frac{2023}{1024000}}=\frac{1024}{2023}$