Question

Randomly pair 4 keys {a, b, c, d} with 3 locks {a, b, c}. What is P(at least one match)?

Answer 1

Total number of ways to pair 3 locks with 4 keys is computed here as:
= 4*3*2 = 24 ways.

P( at least one match) is computed here as:

= P(all match) + P(2 matches) + P(1 match exactly)

= [1 + (Number of ways to select the 2 locks that will match * Number of ways to select a key for the third lock that wont match) + (Number of ways to select one of the three locks that will match * Number of ways to select from the three remaining keys for 2 locks so that none of those 2 match )] / 24

= [ 1 + 3 + k ]/ 24

k we would be looking manually here as:
{ aa, bc, cb or aa, bc, db or aa, cb, dc or aa }
That is 3 cases when a is correctly matched.

Similarly there would be 3 when key for lock b is correctly matched.
Similarly there would be 3 when key for lock c is correctly matched.

Therefore the required probabiltiy here is computed as:
= (1 + 3 + 9)/24

= 13/24

Therefore 13/24 is the required probability here.