Respuesta :
Answer:
The probability that she gets all the red ones, given that she gets the fluorescent pink one, is P=0.0035 or 0.35%.
Step-by-step explanation:
Susan grabs four marbles at random.
We have to calculate the probabilities that he picks the 3 red ones, given that she already picked the fluorescent pink.
If it is given that the fluorescent pink is already picked, we are left with three red marbles, four green ones, two yellow ones, and four orange ones. A total of 13 marbles.
The probability that the second marble is red is 3 in 13.
The probability that the third marble is also red is 2 (the red marbles that are left) in 12 (the total amount of marbles left), as there is a picking without replacement.
The probability that the fourth marble is 1 in 11.
Then, the probability that the 3 red marbles are picked, is:
[tex]P=P_1\cdot P_2\cdot P_3=\dfrac{3}{13}\cdot \dfrac{2}{12}\cdot \dfrac{1}{11}=\dfrac{6}{1716}=0.0035[/tex]
Using the hypergeometric distribution, we find that there is a 0.0035 = 0.35% probability that she gets all the red ones, given that she gets the fluorescent pink one.
In this problem, the marbles are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.
Hypergeometric distribution:
The probability of x successes is given by the following formula:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- N is the size of the population.
- n is the size of the sample.
- k is the total number of desired outcomes.
In this problem, considering she gets the fluorescent one, we want the probability that she grabs all 3 reds when she chooses 3 balls from a set of 3 + 4 + 2 + 4 = 13.
Thus, the parameters are: [tex]x = 3, N = 13, n = 3, k = 3[/tex]
The probability is:
[tex]P(X = 3) = h(3,13,3,3) = \frac{C_{3,3}*C_{10,0}}{C_{13,3}} = 0.0035[/tex]
0.0035 = 0.35% probability that she gets all the red ones, given that she gets the fluorescent pink one.
A similar problem is given at https://brainly.com/question/24008577
