To improve the accuracy, a majority vote is taken. What is the probability of an incorrect classification?

Question

Here is the full question:

In supervised learning, an ensemble of classifiers contains $17$ classifiers whose individual decisions are combined to classify a new example. A classifier has a $0.8$ chance of making a correct decision and a $0.2$ chance of making an error. To improve the accuracy, a majority vote is taken. What is the probability of an incorrect classification?

I'm not sure I did this correctly but I got, the probability of incorrect classification is $$.2(1-.2)^{17-1} = .2(.8^{16}) = .005629$$

Answer 1

A majority is achieved if $9$ or more persons agree with an incorrect classification. Presumably, you can consider each person as an independent trial, and hence you have $17$ independent trials with probability $p = .2$ of success, where I call "making an error" a success. This is because we are interested in the number of people who make an error. If we call this $X$, then $X$ follows what kind of distribution?

Take a look at your previous question for a hint.

Solution.

Notice that the problem asks $P(X\geq 9)$; however, using the complement, we have $$P(X\geq 9) =1-P(X\leq 8) = 1-\sum_{k = 0}^8\binom{17}{k}(.2)^k(1-.2)^{17-k} = 0.002581463$$