TEST 1c FALL 09 - 1 NAME ISyE 3770 Test 1c Solutions Fall...

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1 NAME ISyE 3770 — Test 1 c Solutions — Fall 2009 This test is 55 minutes long. You are allowed one cheat sheet. Put your nice, simple answers here... 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
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2 1. If P ( A ) = P ( B ) = P ( C ) = 1 / 4 and A , B , and C are disjoint, then what is P ( A B C )? Solution: P ( A B C ) = P ( A ) + P ( B ) + P ( C ) = 1 4 + 1 4 + 1 4 = 3 4 . 2. Suppose that P (it rains today) = 0 . 7, P (it rains tomorrow) = 0 . 8, and P (it rains either day) = 0 . 9. What’s the probability that it rains neither day? Solution: P (Neither Today nor Tomorrow) = 1 - P (it rains either day) = 1 - 0 . 9 = 0 . 1 . 3. If P ( A ) = 0 . 3, P ( B ) = 0 . 3, and P ( C ) = 0 . 5, and A , B , and C are independent, find the probability that at least one of A , B , and C occur. Solution: There may be a faster way to do this, but let’s do it the first way I can think of . . . . By DeMorgan’s Law and independence, P ( A B C ) = 1 - P ( A B C ) = 1 - P ( ¯ A ¯ B ¯ C ) = 1 - P ( ¯ A ) P ( ¯ B ) P ( ¯ C ) = 1 - (0 . 7)(0 . 7)(0 . 5) = 0 . 755 . 4. If I pick a random real number between 0 and 1, what is the probability that the selection will be a number of the form 1 /k , where k = 1 , 2 , . . . ? Solution: 0.
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3 5. TRUE or FALSE? If (i) P ( A B ) = P ( A ) P ( B ), (ii) P ( A C ) = P ( A ) P ( C ), (iii) P ( B C ) = P ( B ) P ( C ), and (iv) P ( A B C ) = P ( A ) P ( B ) P ( C ), then A , B , and C are all independent of each other.
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