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This problem trains for: AMC-8, AMC-10, SAT-I, GMAT.

Robin Hood has made a prisoner of the cat of the Sheriff of Nottingham. Since the cat must somehow decide on a ransom amount for itself, Will Scarlet has placed 3 blue beads, 4 green beads and 5 purple beads in a bowl and let the cat play with them. If the fourth bead to fall out is blue, then they would ask for the highest ransom. Assuming the cat pushes out any bead at random, what is the probability that the fourth bead to fall out is blue?

(characters adapted from Sir Walter Scott's Ivanhoe)

The sample space is formed by the total number of permutations of 12 beads. The sample space has size:

equation

The favorable events are those in which the 4th bead is blue. The rest of the beads may occupy any of the 11 remaining spaces. The number of ways in which the 11 remaining spaces can be occupied is:

equation

However, the number of favorable events is larger that that, since there are 3 different ways of choosing the blue bead that is in the fixed slot:

equation

Therefore, the number of favorable events is:

equation

and the probability:

equation