Suzan sees a bag of marbles, she grabs a handful at random. she has seen a bag containing four red marbles, four green ones, two white ones, and two purple ones. she grabs eight of them. find the probability of the following event, expressing it as a fraction in lowest terms. she does not have all the red ones.

Total number of marbles = four red + four green + two white + two purple = 12

Number of marbles taken by her = 8

We have to find the probability that she does not get red.
Instead we find the probability of getting red and then we can take the complement of (getting red)

Now probability of getting red is given as:

No. Of favourable outcomes = Choose 8 out of 11= 8 C 11= (11!)/(8! * 3!)
= 165

Now to choose red balls, she will choose 4 red balls from 4 red balls and other 4 from remaining 8.
4 C 4 * 4C 8 = 70

Probability ( getting red) = 70/165

Therefore, probability (not getting red) = 1- (70/165) = 95/165 = 19/33