80-110 The Nature of Mathematical Reasoning

Thursday, July 27 2000

Quiz 6

Name: _______________________________

1. Assume you woke up this morning and had no idea whatsoever about what day of the week it was. It could be any of { Mon, Tue, Wed, Thu, Fri, Sat, Sun }. If P(Mon) stands for "the probability that today is Monday", what is the value of P(Mon)?

2. P(Mon) = 1/7.

3. a) As you know, there's always a quiz on Tuesdays and Thursdays, so what is P( quiz | Tue ) ?
b) Since there is a quiz today, what is the probability that today is Thursday, i.e, P( Thu | quiz ) ?

4. a) P( quiz | Tue ) = 1
b) P( Thu | quiz ) = 1/2.

5. What does the following expression mean in English "P( Sun | ~quiz )", where "~" stands for "not"?

6. The probability that it is Sunday, given that there is not a quiz today.

7. State the three axioms of probability theory.

8. 1) 0 <= P(A) <= 1
2) P(W)=P(T)=1
3) P(A v B) = P(A) + P(B), if A and B are incompatible propositions. (Lecture 7/24/00 and Handout #18)

9. State either the definition of Conditional Probability or Bayes' Theorem.

10. Conditional Probability: P(A | B)= P(A & B) / P(B), where P(B)>0.
Bayes' Theorem: P(A | B) = P(B | A) * P(A) / P(B). (Lecture 7/24/00 and Handout #18)