This is a Bayes Rule question, so there are only two ingredients we need: the prior odds and the likelihood ratio. The prior odds of rolling a one are . Now we need the likelihood ratio:
Assume that, conditional on the true roll of the die, Alice and Bob make independent reports. This wasn't stated explicitly but it's reasonable in the context. Then the likelihood ratio simplifies to
so we can calculate the likelihood ratio separately for Bob and Alice. Bob is easier, so we'll start with him. Let p be the probability that Bob tells the truth. Then,
since Bob tells the truth with probability p, lies with probability , and chooses uniformly at random from the 5 numbers that were not rolled when he lies.
Now we'll calculate the contribution from Alice's report. Let q be the probability that she tells the truth. Then we have
The reasoning here is identical to the denominator for Bob, but I wanted to spell it out explicitly because the next step is more involved. For Alice's denominator, we have
so it remains to calculate . We can do this using the law of total probability:
Finally, we can compute the likelihood ratio contribution for Alice! It is given by
Now, combining the likelihood ratio contributions from Bob and Alice, we obtain
and the posterior odds are simply the product of the likelihood ratio and the prior odds:
What's nice about writing things this way is that it shows that it's the odds of Alice and Bob telling the truth, respectively, that matter for the final solution. Using the values given by the OP: and , so we obtain . Finally we can convert this to a probability:
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