Respuesta :
Answer:
The answer is below
Step-by-step explanation:
Since they are two green balls, x cannot assume value of 0 and 1. The minimum number of red balls must be two since there are only two green balls and we need to select 4 balls
For x = 2 (select two red balls from 4 red balls and 2 green balls from 2 green balls):
P(x = 2) = [tex]\frac{C(4,2)*C(2,2)}{C(6,2)} =\frac{6}{15}[/tex]
For x = 3 (select 3 red balls from 4 red balls and 1 green balls from 2 green balls):
P(x = 3) = [tex]\frac{C(4,3)*C(2,1)}{C(6,2)} =\frac{8}{15}[/tex]
For x = 4 (select 4 red balls from 4 red balls and 0 green balls from 2 green balls):
P(x = 4) = [tex]\frac{C(4,4)*C(2,0)}{C(6,2)} =\frac{1}{15}[/tex]
Expected value = E(x) = ΣxP(x) = (2×6/15) + (3×8/15) + (4×1/15) = 40/15 = 2.67
Variance = Σx²P(x) - [E(x)]² = (2²×6/15) + (3²×8/15) + (4²×1/15) - (40/15)² = 80/225 = 0.36
Standard deviation = √variance = √0.36 = 0.6
Using the hypergeometric distribution, it is found that:
- The expected value is of 2.67.
- The variance is of 0.356.
- The standard deviation is of 0.596.
The marbles are chosen without replacement, hence, the hypergeometric distribution is used to solve this question.
Hypergeometric distribution:
[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:
- 6 marbles, hence [tex]N = 6[/tex]
- 4 red marbles, hence [tex]k = 4[/tex]
- She selects 4 marbles, hence [tex]n = 4[/tex].
The expected value is:
[tex]E(X) = \frac{nk}{N}[/tex]
Hence:
[tex]E(X) = \frac{4(4)}{6} = 2.67[/tex]
The expected value is of 2.67.
The variance is:
[tex]V(X) = \frac{nk(N-k)(N-n)}{N^2(N-1)}[/tex]
Hence:
[tex]V(X) = \frac{4(4)(2)(2)}{6^2(6-1)} = 0.356[/tex]
The standard deviation is the square root of the variance, hence:
[tex]\sqrt{V(X)} = \sqrt{0.356} = 0.596[/tex]
- The variance is of 0.356.
- The standard deviation is of 0.596.
A similar problem is given at https://brainly.com/question/19426305
