An interesting task on machine learning There are 5 programs. Each program is a binary classifier, which classifies letters - "Spam" and "Not spam." All classifiers determine the category correctly in 60% of cases, regardless of other classifiers.
Suppose that there is a new classifier that starts each of the modules and classifies the letter in the way that at least 3 out of 5 would classify it.
What accuracy will such a classifier have?
I assume that this is solved through the Bayesian theorem.
 A: Assuming they really are independent, then each classifier has an independent 40% chance of getting it wrong for any given item. Your new program, if I understand it correctly, picks the majority "opinion" of the five original classifiers. This will be an opinion that is shared between either 3, 4 or 5 classifiers - these are the three scenario's we must consider.
Classifiers that agree must either all be wrong or all be right. There are 10 permutations in which three classifiers get it right, and each of these permutations has a probability of $0.4^2 \times 0.6^3 = 0.0346$. So, the overall probability that you'll base your final decision on three correct classifications is $10 \times 0.0346 = 0.346$.
There are 5 permutations in which four classifiers get it right, each of which has a probability of $0.4 \times 0.6^4 = 0.0518$. The total probability of this scenario occurring is therefore $5 \times 0.0518 = 0.2592$.
Finally, it is possible for all classifiers to get it right, which can only happen in one configuration (only 1 unique way to order five identical answers), which has a probability of $0.6^5 = 0.0778$. 
So the combined probability that your majority-rule decision is correct is $0.346 + 0.2592 + 0.0778 = 0.6826$. 
More generally, the solution to this kind of problem is given by:
$$
p_{correct} = \sum_{k=\frac{n}{2}+1}^{k=n} \binom{n}{k} (1-p)^{n-k} \times p^k
$$
where $n$ is the number of independent classifiers you're basing your decision on and $p$ is the probability that each of these classifiers is correct. Note that Bayes' theorem does not enter into this.
In practice, if the five classifiers are basing their judgments on the same data, then as whuber pointed out, it is very unlikely that they will actually be independent. For that to happen would basically require that each classifier looks at a separate subset of the data (e.g. different chunks of each letter), and that these subsets are also uncorrelated. 
