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Bob and Alice are expecting triplets. Assume that the probability of each child being born a boy is 0.4, and the gender of each of the children is independent of the others. Suppose events A and B are defined as follows. Are A and B independent?
Carry at least 3 decimals in your calculations.

A - Bob and Alice have children of both sexes
B- Bob and Alice have at most one girl (Hint: Use one of the equations for independence to answer the question Calculate the right side, calculate the left side, and see if they are equal)

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Answer:

Events A and B are not independent.

Step-by-step explanation:

In order for two events to be independent, the following relationship must be true:

[tex]P(A\cap B) = P(A)*P(B)[/tex]

The intersection between events A and B is the outcome that Bob and Alice have exactly one girl, the probability is:

[tex]P(A\cap B) = 0.4*0.4*(1-0.4)\\P(A\cap B) = 0.096[/tex]

For event A, the possible outcomes are having one or two girls:

[tex]P(A) = 0.4*0.4*(1-0.4)+0.4*(1-0.4)*(1-0.4)\\P(A) = 0.240[/tex]

For event B, the possible outcomes are having none or one girl:

[tex]P(B) = 0.4*0.4*(1-0.4)+0.4*0.4*0.4\\P(A) = 0.16[/tex]

Therefore, P(A) x P(B) is:

[tex]P(A) *P(B)= 0.240*0.160=0.0384\\[/tex]

Since P(A) x P(B) ≠ P(A∩B), events A and B are not independent

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