MACM 101 Lecture 7: Lecture 7

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13 Aug 2016
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Monty Hall offers you a choice of 3 doors. One contains a prize, but the other two contain nothing. After choosing your
door, Monty reveals a door which does not contain the prize and then offers to let you switch doors.
What should you do?
Solution: the probability you selected the correct door initially is 1/3 so P(wrong door) = 2/3. After Monty eliminates a
door, is it now a 50-50 guess? NO.
Case A (1 in 3): the prize is behind door #1. in this case, you do best to keep the door.
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Case B (2 in 3): the prize is not behind door #1. the best strategy is to switch doors.
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Suppose we selected door #1. There are 2 cases:
So, switching doors yields P(win) = 2/3
Definition: a proposition is a statement that is either true or false, but not both. The truth value is one of true or false,
which is denoted by 1 or 0 in Grimaldi and perhaps T or F in other texts.
E.g. Give examples of propositions. Give examples of statements which are not propositions.
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It is raining in Vancouver.
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2 + 3 = 6
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Propositions:
Can Mr. Bart give everyone F's in this course?
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University students are old.
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Hello
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Come here!
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x + y = 15 [predicate]
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Not Propositions:
We can build bigger and more complex propositions by "modifying" existing propositions.
Negation
Let p: Bill does his MACM homework everynight
Then : Bill doesn't do his MACM homework everynight
The symbol us the negation symbol, it negates p
All other modifications use two existing propositions to construct a third compound proposition.
Conjunction
p: The wagon is red.
q: Elmo is a green monster.
: the wagon is red and Elmo is a green monster (p and q)
P and q must both be true for p^q to be true; otherwise, p^q is false
Disjunction
p: I am in bed.
q: I am watching television.
I am in bed or I'm watching TV. (p or q)
Lecture 7
January 20, 2016
10:15 AM
Lecture Notes Page 1
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p: I am in bed.
q: I am watching television.
I am in bed or I'm watching TV. (p or q)
P v q evaluates to true exactly when at least one of p and q is true, and evaluates to false otherwise.
r: I am going to the bathroom.
When we say the English sentence:
"I am watching TV or I am going to the bathroom" we don't realistically expect me to be doing both of them at the same
time, do we?
Se, we say exclusive or.
: "I am watching TV or I am going to the bathroom, but not both."
(Exclusive or) evaluates to true when exactly one of q and r is true; false if both are true; false if both are false
Implication
p: It is raining outside
q: I raise my umbrella
P --> q : If it is raining outside then I raise my umbrella. (if p then q; p implies q)
P --> q evaluates to true exactly when p is false or q is true; only false if p is true and q is false at the same time
Lecture Notes Page 2
find more resources at oneclass.com
find more resources at oneclass.com