Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, two green ones, three 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. HINT [See Example 1.]
She has at least one green one.
To find the probability of the event "She has at least one green marble," we need to find the probability of the complementary event "She has no green marbles" and then subtract it from 1.
Total number of marbles in the bag = 4 red + 2 green + 3 white + 2 purple = 11 marbles
Total number of possible outcomes when she grabs 8 marbles = 11C8 = 165 (combination formula)
Now, let's calculate the number of outcomes where she has no green marbles:
Number of ways to choose 8 marbles with no green marbles = (4 red + 3 white + 2 purple)C8 = 9C8 = 9 (combination formula)
Therefore, the number of outcomes where she has at least one green marble = Total outcomes - Outcomes with no green marbles = 165 - 9 = 156
Finally, we can calculate the probability of the event "She has at least one green marble" as:
Probability = (Number of outcomes with at least one green marble) / (Total number of possible outcomes)
= 156 / 165
To express this probability as a fraction in lowest terms, we can simplify it by dividing both the numerator and denominator by their greatest common factor (GCF):
GCF(156, 165) = 3
Simplified fraction = (156 / 3) / (165 / 3) = 52 / 55
Therefore, the probability of the event "She has at least one green marble" is 52/55.