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. ```hint Two competing explanations Work out how plausible an all-red sample of this size is under each factory before you bring in how the factory was chosen. ``` ### 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? ### What a Strong Answer Covers - Applying Bayes' theorem with the equal prior, rather than reporting the red rates themselves - The probability of an all-red sample of size $k$ under each factory - A simplified closed form, and a way to evaluate it without overflow or underflow when $k$ is large - Degenerate inputs, such as equal red rates or a factory that makes no red parts ### Follow-up Questions - How does the posterior change if exactly $j$ of the $k$ parts are red? - If factory A supplies a larger share of the parts than factory B, how does the answer change? - When $a$ is greater than $b$, how many consecutive red parts are needed before the posterior for factory A exceeds a target level $c$?