Recently some german federal states had elections. The news used to say that every n-th voter chose party X. While sitting in the train I was thinking about the following. Someone counts all persons in a train and take every n-th person out assuming this person had elected for party X.
Example: Party X was elected by every 8th person and achieves 12,5%. When there are 248 people in a train and the selection is finished 31 people are selected and it is assumed that they are party-X-voters.
But how many of them really voted for X? Can one assume from the numbers above how many people in the set of 31 probably voted for X? My gut feeling tells me that only 12,5% of this set (~4 people) voted for X, but sometimes stochastics has surprising results.
Given probability of being a party-X voter $0.125$, sample population $248$, and selection size $31$, the probability that all the people in the selection are all the party-X voters in the sample is:
$$\underbrace{(0.125)^{31}}_{\substack{\text{everyone in}\\\text{the selection}\\\text{is an X voter}}}\times\underbrace{(0.875)^{248-31}}_{\substack{\text{everyone not in}\\\text{the selection}\\\text{is not an X voter}}} \approx 2.63\times 10^{-41}$$
Remark: Note that the probability of their being exactly $31$ X voters in the sample is:
$$\underbrace{\frac{248!}{31!\,217!}}_{\substack{\text{ways to arrange }\\\text{$31$ voters}\\ \text{among the sample}}}\times(0.125)^{31}\times(0.875)^{248-31} \approx 0.0764$$
What drives the probability down is the chance of all of them being in the selection.