Probability Theory

This assignment is designed to:

• provide practice with some essential probability theory
• help you become more fluent with the terminology and the techniques
• help you grow in confidence in your mathematical abilities

This is a mathematical homework assignment. Show your work. Be clear and concise. Type your assignment.

I strongly recommend you work through these exercises as soon as possible for their instructional content. If you have general questions about the assignment, please post on the Google Group. Finish early, and earn the early bonus.

[10 points: 2 for the first, 3 each for the second and third parts]

Let's begin with the same sample space $\Omega$ we have been using in class: Let $\Omega$ be the set of all outcomes defined by three successive coin flips. Assume that the coin is fair and that the distribution over all singleton events (events containing one outcome) is uniform. Let the event $D$ be the set of all outcomes in which the second flip is a head. Let the event $E$ be the set of all outcomes in which the final flip is a tail.

1. What are $P(D)$, $P(E)$, and $P(D \cap E)$ ?
2. What is $P(D|E)$ ?
3. What is $P(E|D)$ ?

[10 points]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.3)

Compute the probability of the event 'A period occurs after a three-letter word, and this period indicates an abbreviation (not an end-of-sentence marker)'.

• Let is-abbreviation denote the event “this period indicates an abbreviation”, and let three-letter-word denote “a period occurs after a three letter word”
• Assume the following probabilities:
  • $P($is-abbreviation$ | $three-letter-word$) = 0.8$
  • $P($three-letter-word$) = 0.0003$

[60 points; 20 per sub‐problem]

(Adapted from: Manning & Schuetze, p. 59, exercise 2.1)

Use the Set Theory Identities and Axioms of Probability Theory to prove each of the following five statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply good proof technique: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom. See the proofs on the example proofs page.

1. $P(B - A) = P(B) - P(A \cap B)$
   • Note that inside the $P(\cdot)$, the '$-$' operator indicates set difference.
2. $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule)
   • Hint: use the theorem in part #1 as a step in your proof
3. $P(\overline{A}) = 1 - P(A)$
   • Hint: use the theorem in part #1 as a step in your proof

[10 points]

1. How many possible ways can you completely factor the joint probability of six events $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$
2. Apply the chain rule to completely factor the joint probability $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ in one way.

[10 points]

Apply the definition of conditional probability to prove Bayes’ Law. Use the same standard of proof as in problem #1 above.

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