Probability

Question 1

What is the relation between E(X) and E(E(X given Y))?

Answer 1

E(E(X given Y)) = E(X given Y=1) P(Y=1) + E(X given Y=2) P(Y=2) + E(X given Y=3) P(Y=3) = E(X)

Therefore E(E(X given Y)) = E(X)

Question 2

What is P(A U B)

Answer 2

P(A U B) = P(A) + P(B) - P(A N B)

Question 3

Mutually exclusive

Answer 3

Two events cannot happen at the same time: one happening precludes the other from occurring

P(A n B) = 0

Question 4

What is (A U B) for mutually exclusive events?

Answer 4

P(A U B) = P(A) + P(B) - 0

Question 5

What is the compliment of A?

Answer 5

Compliment of A = A’ = not A

Question 6

What is (A N B)’

Answer 6

All area of Venn diagram except the intersection of A&B

(A N B)’ = A’ U B’

Question 7

What is (A U B)’

Answer 7

All area of Venn diagram outside the circles but inside the box still.

(A U B)’ = A’ N B’

Question 8

What is (A’ N B)

Answer 8

Just B = inside the B circle but not including the intersection

Question 9

What is (A’ U B)

Answer 9

= all area except just A

Question 10

What is A U (B N C)

Answer 10

(A U B) N (A U C)

Question 11

What is A N (B U C)

Answer 11

(A N B) U (A N C)

Question 12

What is P(A U B U C)

Answer 12

P(A U B U C) = P(A) + P(B) + P(C) - P(A N B) - P(A N C) - P(B N C) + P(A N B N C)

Question 13

Combinations definition & formula

Answer 13

nCr = n! / (n - r)! r!

Don’t care about order: AB & BA the same so count as 1

Question 14

Permutations definition& formula

Answer 14

nPr = n! / (n - r)!

Care about order: AB and BA count at 2

Question 15

What is P(A given B) equal to?

Answer 15

P(A given B) = P(A N B) / P(B)

Question 16

Bayes theorem

Answer 16

P(A given B) = P(B given A) P(A) / [P(B given A) P(A) + P(B given A’) P(A’)]

Question 17

Checking for independence of two events

Answer 17

P(A N B) = P(A) P(B) 
since P(A given B) is just P(A) - conditioning has no effect