Probability of Collision in a Hash Table
To understand the answer to this question, let's first understand how the probability of collision in a hash table is calculated.

Given a hash function that distributes keys uniformly, the probability of a collision between two keys is determined by the formula:

P(collision) = 1 - P(no collision)

Where P(no collision) is the probability that a new key hashed does not collide with any existing key in the hash table.

Calculating the Probability of No Collision
In a hash table with a size of 20, the probability of no collision for the first key is 1 (since there are no existing keys).

For the second key, the probability of no collision is 19/20, as it can be hashed to any of the 19 remaining empty slots.

For the third key, the probability of no collision is 18/20, as it can be hashed to any of the 18 remaining empty slots.

Similarly, for the fourth key, the probability of no collision is 17/20.

In general, for the nth key, the probability of no collision is (20 - (n-1))/20.

Finding the Number of Keys for P(collision) > 0.5
We want to find the number of keys after which the probability of collision exceeds 0.5.

So, we need to find the smallest value of n for which P(no collision) < />

Using the formula above, we can calculate the probabilities for different values of n:

P(no collision) for n = 1: 1
P(no collision) for n = 2: 19/20
P(no collision) for n = 3: 18/20
P(no collision) for n = 4: 17/20

Continuing this calculation, we find:

P(no collision) for n = 5: 16/20
P(no collision) for n = 6: 15/20
P(no collision) for n = 7: 14/20

Therefore, for n = 7, the probability of collision exceeds 0.5, as P(no collision) = 14/20 = 0.7.

Conclusion
Based on the calculation above, after hashing 6 keys, the probability that any new key hashed collides with an existing one exceeds 0.5. Therefore, the correct answer is option 'b' - 6.