We have three light bulbs on a shelf. We randomly pick one for testing. We know that among the bulbs there are two relatively non-optimal ('bad') and one relatively optimal ('good') regarding luminosity.
Let $\alpha, \beta$ and $\gamma$ denote the three bulbs, where $\alpha$ is the appropriate bulb.
The corresponding probabilities that a given bulb will produce maximal luminosity per trial are $P_{\alpha}$, $P_{\beta}$, and $P_{\gamma}$, respectively.
We carry out $N$ independent tests for which we observe that the chosen bulb worked well $k$ times.
What's the probability that we picked up the the good one for testing?
My solution:
Let $A$ denote the event that we picked out the appropriate bulb, and let $B$ denote the event that $k$ successes occurred in $N$ trials.
Then the desired conditional probability is the following: \begin{align*} P(A \mid B) &= \frac{P(A \cap B)}{P(B)}\\ &= \frac{\binom{N}{k}\cdot P_{\alpha}^{k} \cdot\left( 1 - P_{\alpha}\right)^{N -k}}{\binom{N}{k}\cdot P_{\alpha}^{k} \cdot\left( 1 - P_{\alpha}\right)^{N -k} + \binom{N}{k}\cdot P_{\beta}^{k} \cdot\left( 1 - P_{\beta}\right)^{N -k} + \binom{N}{k}\cdot P_{\gamma}^{k} \cdot\left( 1 - P_{\gamma}\right)^{N -k}}\\ &= \frac{P_{\alpha}^{k} \cdot\left( 1 - P_{\alpha}\right)^{N -k}}{P_{\alpha}^{k} \cdot\left( 1 - P_{\alpha}\right)^{N -k} + P_{\beta}^{k} \cdot\left( 1 - P_{\beta}\right)^{N -k} + P_{\gamma}^{k} \cdot\left( 1 - P_{\gamma}\right)^{N -k}}. \end{align*}