Exercise 3.18 - Bayes factor for coin tossing

The Bayes factor for hypothesis test is defined by:

where $H_0$ is the null hypothesis.

We have:

And

according to Exercise 3.17. The Bayes factor is approximately $9.3$.

When $N=100$ and $N_1=90$, the Bayes factor is:

When $\frac{N}{N_1}$ remains a constant deviated from 0.5, the larger $N$ is, the more likely that the coin is biased. This is an intuitive conclusion from the law of large numbers.