Question

Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let Xi equal 1 if the ith ball selected is white and let it equal 0 otherwise. Give the joint probability mass function of X1 , X2.

Answer 1

Answer:

The joint probability mass function table is provided below.

Step-by-step explanation:

The experiment consist of drawing 3 balls from an urn containing 5 white and 8 red balls.

The random variable Xi is defined as follows:

Xi = 1; if the ith ball is white

Xi = 0; otherwise.

Then X₁ = the 1st ball is white and X₂ = the 2nd ball is white.

The sample space is as follows:

S = {(X₁ = 0, X₂ = 0), (X₁ = 0, X₂ = 1), (X₁ = 1, X₂ = 0), (X₁ = 1, X₂ = 1)}

Compute the probability of each event in the sample space as follows:

$P(X_{1} = 0, X_{2}= 0)=\frac{8}{13} \times\frac{7}{12}=\frac{14}{39}$

$P(X_{1} = 0, X_{2}= 1)=\frac{8}{13} \times\frac{5}{12}=\frac{10}{39}$

$P(X_{1} = 1, X_{2}= 0)=\frac{5}{13} \times\frac{8}{12}=\frac{10}{39}$

$P(X_{1} = 1, X_{2}= 1)=\frac{5}{13} \times\frac{4}{12}=\frac{5}{39}$

The joint probability mass function table is provided below.