A bag contains 7 red marbles, 5 white marbles, and 6 blue marbles. You draw 4 marbles out at random, without replacement. What is the probability that exactly two of the marbles are red?

Answer 1

There are a number of ways to determine the sample space including a tree diagram but the bottom line is this:

Case        Prob     Prob    Prob    Prob    Total

RRWW   7/18      6/17    5/16    4/15    840/73440

RRWB    7/18      6/17    5/16    6/15    1260/73440

RRBW    7/18      6/17    6/16    5/15    1260/73440

RRBB     7/18      6/17    6/16    5/15    260/73440

RWRW   7/18      5/17    6/16    4/15    840/73440

RWRB    7/18      5/17    6/16    6/15    1260/73440

RWWR   7/18      5/17    4/16    6/15    840/73440

RWBR    7/18      5/17    6/16    6/15    1260/73440

RBRW    7/18      6/17    6/16    5/15    1260/73440

RBRB     7/18      6/17    6/16    6/15    1512/73440

RBWR    7/18      6/17    5/16    6/15    1260/73440

RBBR     7/18      6/17    5/16    6/15    1260/73440

WRRW   5/18      7/17    6/16    4/15    840/73440

WRRB    5/18      7/17    6/16    6/15    1260/73440

WRWR   5/18      7/17    4/16    6/15    840/73440

WRBR    5/18      7/17    6/16    6/15    1260/73440

WWRR   5/18      4/17    7/16    6/15    840/73440

WBRR    5/18      6/17    7/16    6/15    1260/73440

BRRW    6/18      7/17    6/16    5/15    1260/73440

BRRB     6/18      7/17    6/16    5/15    1260/73440

BRWR    6/18      7/17    5/16    6/15    1260/73440

BRBR     6/18      7/17    5/16    6/15    1260/73440

BWRR    6/18      5/17    7/16    6/15    1260/73440

BBRR     6/18      5/17    7/16    6/15    1260/73440

                                                     Total   27972/73440

Each line is a different combination of choices that yields exactly 2 reds. Determining the probability of each and then summing them up results in 27972/73440 which can be reduced to 259/680