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 one purple one. 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 does not have all the red ones.

Probability of picking 8 marbles out of 10 where the 8 marbles do not contain all of the red marbles = (2/3)

Step-by-step explanation:

In the bag, there are four red marbles, two green ones, three white ones, and one purple one.

In total, there are (4+2+3+1) = 10 marbles

Suzan grabs 8 of them. Find the probability that she doesn't have all the red ones.

(The probaility that she doesn't have all the red marbles) =

1 - (Probability that she has all the red marbles)

Since, we are working with a subset of only red marbles and others, we can say there are four red marbles and 6 other marbles.

To find the probability that she has all the red marbles among the 8 picked marbles,

Using the law of probability that the probability of an event = (Number of elements in the event) ÷ (Number of elements in the sample space)

Number of elements in the sample space = n(S) = how many different ways 8 marbles can be picked from 10 marbles = ¹⁰C₈ = 45

Number of elements in the event = n(E) = how many ways 8 marbles can be picked such that 4 of the 8 marbles are red amd the other 4 are of other colours.

Ways of picking 4 marbles out of 4 = ⁴C₄ = 1

ways of picking 4 marbles from 6 other marbles = ⁶C₄ = 15

Ways of picking 4 red marbles out of 4 red marbles and picking 4 other-colour marbles from 6 other-colour marbles = 1 × 15 = 15 ways

Probabilty of picking 8 marbles out of 10 where the 8 marbles contain all of the red marbles = (15/45) = (1/3)

Probability of picking 8 marbles out of 10 where the 8 marbles do not contain all of rhe red marbles = 1 - (1/3) = (2/3)

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