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Problem Set 7

MATH-UA 120 Discrete Mathematics

due December 1, 2023

These are to be written up in LATEX and turned in to Gradescope.

LATEX Instructions: You can view the source (.tex) file to get some more  examples of LATEX code. I have commented the source file in places where new LATEX constructions are used.

Remember to change \showsolutionsfalse to \showsolutionstrue in the document’s preamble (between \documentclass{article} and \begin{document})

Assigned Problems

1. A fair coin is flipped 10 times.

(a) What is the probability that there are an equal number of head and tails?

(b) What is the probability the first three flips are heads?

(c) What is the probability that there are an equal number of heads and tails and the first three flips are heads?

(d) What is the probability that there are an equal number of heads and tails or the first three flips are heads (or both)?

(e) What is the probability that the first three flips are heads given that an equal number of heads and tails are flipped?

2. An unfair coin shows heads with probability p and tails with probability 1 − p. Suppose this coin is flipped 2 times. Let A be the event that the coin comes up first heads and then tails. Let B be the event that the coin comes up first tails and then heads.

(a) Find P(A).

(b) Find P(B).

(c) Find P(A | A ∪ B).

(d) Find P(B | A ∪ B).

Explain how one could use this unfair coin to make a fair decision.

3. Suppose that A and B are events in a sample space (S, P). Prove or disprove:

(a) If P(A ∩ B) = 0, then P(A | B) = P(B | A) if and only if P(A) = P(B).

(b) If P(A) > 0, P(B) > 0 but P(A ∩ B) = 0, then P(A | B) = P(B | A). If proven, give an example of two such events with P(A) ≠ P(B).

4. Consider the sample space S = {a, b, c}, with equal probability for each outcome. Define the random variables X and Y by X(a) = −1, X(b) = 0, X(c) = 1, Y (a) = Y (c) = 0, Y (b) = 1. Check that Var(X + Y ) = Var(X) + Var(Y ), but that X and Y are not independent.

5. Provide an alternative proof to Proposition 31.7 using any of the statements in Proposition 31.8.

Theorem (Proposition 31.7). Let A and B be events in a sample space (S, P).

Then

P(A) + P(B) = P(A ∪ B) + P(A ∩ B).