Let's actually calculate that probability.

We started with two hypotheses, good die is on the right or the bad die

is on the right.

And we said that initially, we're going to give these equal chances of

50% chance of being true before we

actually get started with the data collection.

Remember these were our priors.

Then, we think about the data collection stage.

If it is true that the good die is on the right, the probability

of rolling a number greater than or equal to 4 is going to be 75%.

And the

complement of that, rolling a number less than 4, is going to be 25%.

If, on the other hand, you're actually holding the bad

die on the right, and you're picking the right hand.

The probability of rolling a number greater than or equal to 4 is

only 50%, and the compliment, rolling a number less than 4, is also 50%.

Usually in probability trees, the next

step is to calculate the joint probabilities.

So we multiply across the branches, there's a 37.5% chance that

the good die is on the right and you roll a number greater than or equal to 4.

There is a 12.5% chance that the good die is

on the right and you roll a number less than 4.

There is a 25% chance that the bad die is on the right and you roll a number greater

than or equal to 4.

And there's a 25% chance that the bad die is

on the right and you roll a number less than 4.

Remember, we did indeed roll a number greater than or equal to 4.

So these are the two outcomes that we're

most interested in, the very top branch and the

third branch, good die on the right and roll a number greater than or equal to 4.

Or, bad die on the right and roll a number greater than or equal to 4.