call from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, two green ones, four 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 does not have all the red ones

3 red

2 green
4 white
2 purple ---> total = 11

let's do the prob that she had the 3 reds
number of such choices = C(3,3) x C(8,5) = 56
number of choices of 8 from 11 = C(11,8) = 165

prob(all 3 reds) = 56/165

so prob(not all the reds) = 1 - 56/165 =109/165

To find the probability of the event "Suzan does not have all the red marbles," we need to calculate the ratio of favorable outcomes to total outcomes.

First, let's determine the total number of possible outcomes. Suzan grabs eight marbles from the bag, so the total number of possible outcomes can be calculated using combinations. We can express this as C(11, 8), where 11 represents the total number of marbles in the bag (3 red + 2 green + 4 white + 2 purple ones) and 8 represents the number of marbles Suzan grabs.

C(n, r) is calculated using the formula: n! / ((n - r)! * r!), where "!" denotes the factorial operation.

C(11, 8) = 11! / ((11 - 8)! * 8!)
= 11! / (3! * 8!)
= (11 * 10 * 9) / 3!

Next, let's determine the number of favorable outcomes, which is the number of outcomes where Suzan does not have all the red marbles.

To calculate this, we need to consider two cases:
1. Suzan grabs no red marbles.
2. Suzan grabs some red marbles (at least one), but not all.

Case 1: Suzan grabs no red marbles.
In this case, Suzan can only choose from the remaining marbles (2 green + 4 white + 2 purple ones). She needs to select 8 marbles, so the number of favorable outcomes is C(8, 8).

Case 2: Suzan grabs some red marbles (at least one), but not all.
In this case, Suzan can choose from the remaining red marbles (3 - 1 = 2), as well as the other marbles (2 green + 4 white + 2 purple ones). She needs to select 8 marbles, so the number of favorable outcomes is C(2, 1) * C(9, 7).

Using the same combination formula as before, we can calculate the values for these cases:
C(8, 8) = 8! / ((8 - 8)! * 8!) = 1
C(2, 1) * C(9, 7) = (2! / (2 - 1)! * 1!) * (9! / ((9 - 7)! * 7!)) = 2 * 36

Now, we can calculate the total number of favorable outcomes:
Number of favorable outcomes = C(8, 8) + C(2, 1) * C(9, 7) = 1 + 2 * 36 = 73

Finally, we can calculate the probability as the ratio of favorable outcomes to total outcomes:
Probability = Number of favorable outcomes / Total number of outcomes = 73 / [(11 * 10 * 9) / 3!]

To express the probability as a fraction in lowest terms, we simplify the expression:
Probability = 73 / [(11 * 10 * 9) / 3!] = 73 / 165

Therefore, the probability of Suzan not having all the red marbles is 73/165.

To find the probability of the event that Suzan does not have all the red marbles, we can use the concept of complementary probability.

First, let's determine the total number of possible outcomes. Suzan chooses 8 marbles from a bag containing 3 red marbles, 2 green ones, 4 white ones, and 2 purple ones. Therefore, there are a total of 8 marbles to choose from.

Next, let's find the number of outcomes where Suzan does have all the red marbles. In this case, she would have to choose all 3 red marbles from the bag. The number of ways she can do this is given by the combination formula:

C(3, 3) = 1

Now, we can find the number of outcomes where Suzan does not have all the red marbles. This is equal to the total number of outcomes minus the number of outcomes where she does have all the red marbles:

Total outcomes - Outcomes with all red marbles = 8 - 1 = 7

Therefore, the probability of the event that Suzan does not have all the red marbles is given by:

P(not all red marbles) = (number of outcomes without all red marbles) / (total number of outcomes)
= 7 / 8

Since the probability should be expressed as a fraction in its lowest terms, the final answer is:

P(not all red marbles) = 7/8