Bayesian Posterior That k Red Parts Came From Factory A

Two factories make the same part in two colors. Factory A's output is $a%$ red and $(100-a)%$ black. Factory B's output is $b%$ red and $(100-b)%$ black. You pick one of the two factories at random, each with equal probability, and draw $k$ parts from its output. All $k$ parts turn out to be red.

What is the conditional probability that these $k$ parts came from factory A? Give a general expression in $a$, $b$ and $k$, then evaluate it for the values you are given. The assessment generates the specific numbers randomly, and each question states whether the answer is an integer or a decimal rounded to two places.

Constraints and Clarifications

• $a$ and $b$ are percentages between 0 and 100, and $k$ is a positive integer.
• Each factory is chosen with probability one half, and all $k$ parts come from the chosen factory.
• Treat each factory's output as very large, so the colors of the drawn parts are independent given the factory.

Clarifying Questions

• Is each factory equally likely to be chosen, or does one of them supply more of the parts?
• Are the parts drawn from a large production run, or from a small batch of known size where drawing without replacement matters?
• Should rounding happen only at the end, and to how many decimal places?