Final answer:
The probability of a simple majority voting ensemble model making an incorrect prediction can be calculated using the binomial probability formula. For ensembles with 11 models, the probability is the sum of the probabilities from 6 to 11 models being wrong. For an ensemble with 21 models, it's the sum from 11 to 21 models being wrong. Exact calculations usually require computational tools.
Step-by-step explanation:
To calculate the probability of a model ensemble using simple majority voting making an incorrect prediction, we can use the binomial probability formula. The binomial probability of exactly k successes in n independent Bernoulli trials, each with success probability p, is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where C(n, k) is the number of combinations of n items taken k at a time.
For majority voting to fail, more than half of the models must make incorrect predictions. For 11 models (a and b), this is 6 or more models making an incorrect prediction. For 21 models (c), this is 11 or more incorrect predictions.
The probability of an incorrect prediction by the majority voting of 11 models with a 0.2 error rate (a) and 11 models with a 0.49 error rate (b) can be calculated by summing the probabilities of 6 through 11 models being incorrect. For 21 models with a 0.49 error rate (c), we sum the probabilities of 11 through 21 models being incorrect.
As calculating these probabilities involves a sum of several terms involving combinations and powers, the exact values typically require computational assistance, such as using a statistical software package or programmable calculator.