Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, three green ones, four white ones, and three purple ones. She grabs seven of them. Find the probability of the following event, expressing it as a fraction in lowest terms.
She has all the red ones.
So it should be that n(s)=c(13,7)=? I don't know how to get the answer to what that is equal to and also after that, it should be n(e)=c(3,3)*c(7,4)=?
You are partly on the right track.
The C(13,7) is the number of to choose 7 of the 13 marbles.
you want the case of 3 reds plus 4 non-reds
prob(your stated event) = C(3,3)*C(10,4)/C(13,7)
= 1*210/1716
= 35/286
To find the probability of Suzan grabbing all the red marbles, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
1. Finding the number of total outcomes:
The total number of outcomes is the number of ways Suzan can select 7 marbles from the bag without any restrictions. This can be calculated using the combination formula:
n(s) = C(13, 7)
The combination formula is calculated as:
C(n, r) = n! / (r!(n - r)!)
So, in this case:
n(s) = C(13, 7) = 13! / (7!(13 - 7)!) = 13! / (7!6!)
2. Finding the number of favorable outcomes:
Since Suzan wants all the red marbles, she needs to select all 3 red marbles and any other 4 marbles from the remaining 10 (3 + 10 = 13).
First, calculate the number of ways to select the 3 red marbles:
C(3, 3) = 1 (there is only one way to select all 3 marbles out of 3)
Then, calculate the number of ways to select the remaining 4 marbles from the remaining 10:
C(10, 4)
Multiply these two calculations to find the number of favorable outcomes:
n(e) = C(3, 3) * C(10, 4) = 1 * C(10, 4)
3. Simplify n(s) and n(e) if possible to express them as fractions in lowest terms.
Finally, the probability of the event (Suzan grabbing all the red marbles) can be found by dividing the number of favorable outcomes by the total number of outcomes:
P(e) = n(e) / n(s) = (C(3, 3) * C(10, 4)) / (C(13, 7))
To find the probability of Suzan grabbing all the red marbles, we need to determine the total number of possible outcomes (n(s)) and the number of favorable outcomes (n(e)).
The total number of possible outcomes (n(s)) is the number of ways Suzan can choose seven marbles from the bag, without any restrictions. We can find this using combinations. The formula for combinations is:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of items and r is the number of items chosen. In this case, n = 13 (the total number of marbles in the bag) and r = 7 (the number of marbles she grabs).
So, n(s) = C(13, 7) = 13! / (7! * (13 - 7)!) = 1716
Next, we need to determine the number of favorable outcomes (n(e)), which is the number of ways Suzan can choose three red marbles out of seven chosen overall.
Since there are three red marbles in the bag, Suzan needs to pick three red marbles out of seven total marbles chosen. We can find the number of ways to do this by calculating the combination of choosing three red marbles from seven chosen marbles (C(3, 3)), multiplied by the number of ways to choose the remaining four marbles from the non-red marbles (C(7, 4)).
n(e) = C(3, 3) * C(7, 4) = 1 * 35 = 35
Now, we can find the probability of the event by dividing the number of favorable outcomes by the total number of possible outcomes:
P(E) = n(e) / n(s) = 35 / 1716
This fraction cannot be simplified further, so the probability of Suzan grabbing all the red marbles is 35/1716.