Suzy has a bag containing 5 red marbles, 2 green ones, 4 white ones, and 3 purple ones. She grabs 9 of them. Find the probability that she does not have all the green ones. Express the answer as a fraction in lowest terms.
Find the prob of getting both green ones
which is
C(2,2)*C(12,7)/C(14,9)
= 792/2002
so prob(not all green) = 1- 792/2002
= 1210/2002
55/91
To find the probability that Suzy does not have all the green marbles, we need to calculate the number of favorable outcomes (Suzy not having all the green marbles) divided by the total number of possible outcomes (Suzy selecting 9 marbles from the bag).
The total number of marbles in the bag is 5 (red) + 2 (green) + 4 (white) + 3 (purple) = 14 marbles.
To calculate the number of favorable outcomes, we consider two cases:
Case 1: Suzy selects no green marbles. In this case, she can select 9 marbles from the remaining 14 - 2 = 12 marbles (excluding the green marbles). This can be calculated as 12 choose 9.
Case 2: Suzy selects at least one green marble, but not all the green marbles. In this case, we have 3 possibilities: choosing 1 green marble + selecting 8 marbles from the remaining 14 - 1 = 13 marbles (excluding the one green marble), or choosing 2 green marbles + selecting 7 marbles from the remaining 14 - 2 = 12 marbles, or choosing 3 green marbles + selecting 6 marbles from the remaining 14 - 3 = 11 marbles. This can be calculated as (2 choose 1) * (12 choose 8) + (2 choose 2) * (12 choose 7) + (2 choose 3) * (11 choose 6).
The total number of possible outcomes is 14 choose 9.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability (not having all the green marbles) = [Case 1 + Case 2] / Total
= [(12 choose 9) + [(2 choose 1) * (12 choose 8) + (2 choose 2) * (12 choose 7) + (2 choose 3) * (11 choose 6)]] / (14 choose 9)
Calculating these combinations, we get:
[220 + (2 * 495 + 1 * 792 + 0 * 462)] / 2002
= (220 + 990 + 792) / 2002
= 2002 / 2002
= 1
Therefore, the probability that Suzy does not have all the green marbles is 1, which can be expressed as the fraction 1/1 in lowest terms.