Mathematics: If 3 balls are drawn at random from a bag containing 3 red and 4 blue balls, what is the expected number of red balls in the sample?

The answer is A,C,D Step-by-step explanation:...

If 3 balls are drawn at random from a bag containing

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18.06.2021

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:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

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:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The expected value of the hypergeometric distribution is:

$E(X) = \frac{nk}{N}$

3 red balls in the sample:

This means that k = 3

3 balls are drawn:

This means that n = 3

Total of 3 + 4 = 7 balls:

This means that N = 7

What is the expected number of red balls in the sample?

$E(X) = \frac{nk}{N} = \frac{3*3}{7} = 1.2857$

The expected number of red balls in the sample is 1.2857.

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