Answer:
The expected number of red balls in the sample is 1.2857.
Step-by-step explanation:
The balls are chosen without replacement from the sample, which means that the hypergeometric distribution is used to solve this question.
Hypergeometric distribution:
The probability of x sucesses 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]
In which:
x is the number of sucesses.
N is the size of the population.
n is the size of the sample.
k is the total number of desired outcomes.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The expected value of the hypergeometric distribution is:
[tex]E(X) = \frac{nk}{N}[/tex]
3 red balls in the sample:
This means that [tex]k = 3[/tex]
3 balls are drawn:
This means that [tex]n = 3[/tex]
Total of 3 + 4 = 7 balls:
This means that [tex]N = 7[/tex]
What is the expected number of red balls in the sample?
[tex]E(X) = \frac{nk}{N} = \frac{3*3}{7} = 1.2857[/tex]
The expected number of red balls in the sample is 1.2857.
