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Probability : Independent Events and Hemophilia

1.(a) Let. A, B, C are mutually independent. Prove that A. is conditionally independent of B given C.
(b) Assume that A, B are both independent and conditionally independent given C. Is it necessary that A, B, C are mutually independent?

2. Hemophilia is a hereditary disease. If a mother has it, then with probability 1/2, any of her sons independently will inherit it. Otherwise, none of the sons will become heniophilic. Julie is a mother of two sons and from her medical history it is known that, with probability 1/4, she is hemophilic. That is the probability that
(a) her first son is hemophilic;
(h) her second son is hemophilic;
(c) both sons are hemophilic?
Cards are drawn at random from an ordinary deck of 52 cards, one- by-one without replacement. What is the probability that no heart is drawn before the ace of spades is drawn?

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Problem #1
(a) Since , and are mutually independent, then we have
, ,
Thus we have

Thus is conditionally independent of given .
(b) No.

Problem #2
Let denote the event that Julie has ...

Solution Summary

Independent Events and Hemophilia are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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