Discrete Math Assignment Help

PRACTICE PROBLEMS FOR THE FINAL EXAM

Here are some sample problems for the Final Exam. These do not necessarily reflect

the difficulty, length, format, or sum total of all material for the exam.

(1) Are the following logical propositions equivalent: ¬((P ∧Q) ⇒ R) and P ∧Q∧(¬R).

(2) Prove the following statements or demonstrate that they are false:

(a) There is one and only one minimal spanning tree in every weighted graph.

(b) Every prime number p > 2 is odd.

(c) For m and n any two negative integers, gcd(m, n) > m, n.

(3) The game of (one pile) NIM proceeds as follows: a pile of 20 objects (rocks, sticks,

whatever) is formed. Two players take turns grabbing objects from the pile. On

their turn, a player must grab either 1, 2, or 3 objects. The player who grabs the

last objects from the pile wins.

For the purposes of this algorithm, we need the following additional algorithm

language: we will use the function PlayerInput to indicate we are requesting input

from a human player and so p ←PlayerInput will assign the requested player input

to the variable p.

Consider the following algorithm which simulates playing NIM against a human

player:

Algorithm NIM winner

n ← 20

repeat until n = 0

p ←PlayerInput

n ← n − p and take p objects from the pile [human turn]

if p = 1 then n ← n − 3 and take 3 objects from the pile [algorithm’s turn]

if p = 2 then n ← n − 2 and take 2 objects from the pile

if p = 3 then n ← n − 1 and take 1 object from the pile

endrepeat

(a) Write the trace where you play the part of the human playing against the algorithm (as part of this trace, you may pick any number of objects you want each

time).

(b) Prove, using the language of loop invariants, that the algorithm will always win

NIM against any human player.

(c) Draw a game tree for two players starting with 4 objects to prove that, with

optimal play, the second player always wins.

Date: May 2, 2021.

(4) Prove the following identity two ways

Xn

k=0

(2k − 1) = n

2 − 1

once directly via induction and once using the language of sequences and unique

determination.

(5) For

A =

1 2 1 0

0 0 0 1

0 0 0 0

1 1 1 0

(a) Draw a graph with A as adjacency matrix.

(b) Find the number of length 3 paths from v1 to v2.

(c) Perform all iterations of Warshall’s algorithm to find the path matrix P for the

given graph.

(6) Find the general solution to the difference equation with characteristic equation (r −

3)3

(r − 2)(r 1) = 0.

(7) In the game Blackjack, the 52 cards in a standard deck of cards are valued as follows:

for the numbered cards 2 through 10, the value matches the value on the card. The

face cards (Jacks, Queens, and Kings) are valued 10. Aces are valued 11 (technically,

they can take a value of 11 or 1 at the choice of the player, but ignore this wrinkle

for the purpose of this problem).

(a) Find the expected value of a single card dealt to you from a full deck of cards

with respect to the above point values.

(b) During play, a player will receive a hand of two cards without replacement. A

“Blackjack” is a hand with an ace and another card with value 10. Find the

probability of receiving a Blackjack, given your first card was an ace.

(c) Find the overall probability of receiving a Blackjack overall.

(8) You watch the weather forecast daily for 100 days and record the outcome of rain

versus no rain, both in terms of what was forecast and what actually occurred that

day. You found that, when rain is forecast, it actually rains 95% of the time and

when no rain is forecast, rains 3% of the time.

(a) Assuming that rain is forecast on 40% of days, what proportion of days did it

actually rain?

(b) Assuming you pick a day out of the 100 days you recorded weather forecasts

and outcomes at random and that day it rained, what was the probability the

forecast that day called for rain?

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