Let’s analyze the scenario:

• Random Classifications: If the classifications of the examples are truly random, there’s no underlying pattern or relationship between the attributes and the class labels.
• Consistent Hypothesis: A consistent hypothesis perfectly predicts the class label for all the observed examples.

Finding a Contradiction:

A contradiction arises when a new example is observed whose classification cannot be accurately predicted by the existing hypothesis. In the case of random classifications, finding a contradiction is highly likely as the number of examples increases.

Calculation:

1. Total Possible Classifications: With ‘n’ Boolean attributes, there are 2^n possible combinations of attribute values. For each combination, there are 2 possible classifications (positive or negative). Therefore, the total number of possible classifications for all examples is 2^(2^n).

2. Probability of No Contradiction:

   • Initially, with a single example, there’s no contradiction.
   • With two examples, the probability of both having the same random classification is 1/2^(2^n).
   • With three examples, the probability of all having the same classification is (1/2^(2^n))^2.
   • For ‘m’ examples, the probability of no contradiction is (1/2^(2^n))^(m-1).

3. Probability of Finding a Contradiction:

   • The probability of finding at least one contradiction is 1 – (probability of no contradiction).

To find the number of examples required for the probability of finding a contradiction to reach 0.5:

1. We need to solve the following equation: 1 – (1/2^(2^n))^(m-1) = 0.5

2. Rearranging the equation: (1/2^(2^n))^(m-1) = 0.5

3. Taking the logarithm of both sides: (m-1) * log(1/2^(2^n)) = log(0.5)

4. Solving for ‘m’: m = 1 + log(0.5) / log(1/2^(2^n))

This equation gives you the number of examples (‘m’) required to reach a 0.5 probability of finding a contradiction.