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Section 9.2

Section: 
9.2
Date: 
Friday, February 25, 2011 - 12:00

Today, we did examples from Section 9.2 in the text. Keep in mind that I am writing the probability trees from top to bottom instead of left to right.

Starting Probability (Section 9.1)

Section: 
9.1
Date: 
Monday, February 21, 2011 - 12:00 - Wednesday, February 23, 2011 - 13:00

Today, we started section 9.1. I passed out the attached PDF, which condenses the book's presentation a bit and helps organize some of the definitions.

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MATH1120–003 Spring 2011 University of Colorado

Section 9.1 Notes

Definitions (by order as found in text)

  • experiment – An activity whose results can be observed and recorded. (e.g., flipping a coin two times)

  • outcome – A possible result of an experiment. (e.g., “heads” in a single coin flip)

  • sample space – A set, usually S, of all possible outcomes for an experiment. (e.g., \{ HH,TT,HT,TH\})

  • event – Any subset of a sample space. (e.g., \{ HH,TT\})

  • empirical probability – When a probability is determined by viewing the results of experiments.

  • theoretical probability – When a probability is determined mathematically, without experimentation.

  • P(E) – The probability of outcome/event E. We always have 0\le P(E)\le 1 for all events E.

  • equally likely – When one outcome is just as likely as another (when their probabilities are the same).

  • n(E) – The number of ways in which the event E occurs in the sample space S.

  • uniform sample space – Each possible outcome in the sample space is equally likely.

  • impossible event – An event E that never occurs as an outcome in a sample space. (P(E)=0)

  • certain event – An event E that is certain to occur, no matter what. (P(E)=1)

  • \emptyset – The set with no elements, known as the empty set.

  • mutually exclusive – Events A and B are mutually exclusive if they have no elements in common.

  • complement – For an event E, the complement \overline{E} is everything in the sample space that is not in E.

Theorems and Their Uses (by order as found in text)

9.1 – Law of Large Numbers (Bernoulli's Theorem)

If an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches a fixed number as the number of repetitions increases.

Idea: Probability is best measured when the experiments in question are repeated, their results being averaged.

Example: If you flip a coin three times, you may have heads land up each time and be lead to think that your coin will land on heads more often. You'd probably be wrong in thinking such a thing, and doing the experiment over and over will show you this.

9.2 –

If A is any event and S is the sample space, then 0\le P(A)\le 1.

Idea: Impossible events have a probability of 0, while certain events have a probability of 1. All other events will have probabilities between 0 and 1.

9.3 –

The probability of an event is equal to the sum of the probabilities of the disjoint outcomes making up the event.

Idea: If you roll a fair die and want to know the probability of rolling an odd number, you could roll a 1, a 3 or a 5. Consider the probabilities for each case individually and sum those:

P(1)+P(3)+P(5)=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}.
9.4 –

If A and B are mutually exclusive, then P(A\text{ or }B)=P(A\cup B)=P(A)+P(B)

Idea: See the previous example.

9.5 –

If A is an event and \overline{A} is its complement, then

P(A)+P(\overline{A})=1

Idea: The above equation can be rewritten in a number of ways, depending on specifically what one is looking for. The main idea is that an event A either happens, or it does not happen. Considering those two as outcomes (A happens, and A doesn't happen) yields a certain event (one of them happens).

Calculating Probabilities

  • Probability of an Event with Equally Likely Outcomes – For an experiment with sample space S with equally likely outcomes, the probability of an event A is given by

    P(A)=\frac{n(A)}{n(S)}.

    The main idea here is to take the total number of ways that the outcome A occurs, and divide it by the total number of outcomes.

  • P(\emptyset)=0 – An impossible event never happens.

  • P(S)=1 – At least one event in S occurs.

  • 0\le P(A)\le 1 – For any event A, probability of A is always between 0 and 1.

  • Union of events – If A and B are events, then P(A\cup B)=P(A)+P(B)-P(A\cap B). (explained in class)

  • Union of mutually exclusive events – If A and B are mutually exclusive, then P(A\cup B)=P(A)+P(B).

  • Complement – If A is any event, then P(\overline{A})=1-P(A).

Assignment 4

Due Date: 
Monday, February 28, 2011 - 16:00

Section 9.1 (pp. 528-533)
A: 1, 3, 5, 8, 13, 15, 16.
B: 2, 4, 6, 9, 12.
Mathematical Connections: 1.

Section 9.2 (pp. 549--554)
A: 1, 2, 3, 5, 6, 8, 9, 13, 16.
B: 3, 5, 6, 11, 14, 15.
Mathematical Connections: 3.

Solutions to Review for Test 1

The solutions to the review for test 1 are attached to this post. We will be covering these questions on Wednesday in class. Bring any questions you may have!!

Review for Test 1

Math 1120 -- Review for the First -- Exam Spring 2011

  1. Express each of the following fractions in simplest form.

    1. \displaystyle{\frac{30}{315}}

    2. \displaystyle{\frac{98}{-63}}

    3. \displaystyle{\frac{627}{704}}

    4. \displaystyle{\frac{-1230}{-3888}}

    5. \displaystyle{\frac{126}{96}}

    6. \displaystyle{\frac{35^{{10}}}{21^{{11}}}}

    7. \displaystyle{\frac{101101}{539}}

In Class, Monday, 2-07-11

Section: 
7.1-7.4
Date: 
Monday, February 7, 2011 - 12:00 - 12:50

Today, we mostly covered homework related questions.

Remember that there is a test next week. There is now a review posted on the website aimed at helping you prepare for the test. The test will cover up through section 8.2. We'll be covering sections 8.1 and 8.2 this week and you should note that there is an assignment on these sections due next week on Monday. You have half as much time (one week instead of two), but the assignment is half the size of the previous one.

Any group that has not sent me their expected completion date for their project should do so as soon as possible.

Assignment 3

Due Date: 
Monday, February 14, 2011 - 16:00

Section 8.1 (pp. 485-489)
A: 1, 2, 3, 5, 6, 8, 9, 12, 15, 18, 21.
B: 1-4, 7, 11, 13, 15, 18.
Mathematical Connections: 5.

Section 8.2 (pp. 500-504)
A: 1-6, 8-10, 15, 22, 24.
B: 1-4, 7, 10, 11, 16, 20, 25, 29.
Mathematical Connections: 8.

The Real Number System

Section: 
7.4
Date: 
Friday, February 4, 2011 - 12:00 - 12:50

Today, we finished topics from Section 7.4 with a discussion of the real number system. Some basic notes:

C - complex numbers (e.g., 2+3i), also called "imaginary numbers"
R - real numbers (all decimals, fractions, natural numbers and integers)
Q - rational numbers (a/b where a and b are integers, b not zero) The Q here stands for "quotient."
Z - integers (e.g., -2, 3, 120, -110, 0). The Z is German for "zahl" = "number".
N - natural numbers (the counting numbers: 1, 2, 3, 4,...)
W - the whole numbers (the natural numbers and zero)

We also discussed some properties of irrational numbers. Here's a weird fact (the proof is left to advanced analysis courses in mathematics): There are infinitely many more irrational numbers than there are rational numbers. But, our minds are wired to think in rational and natural numbers. I asked the class to name as many irrational numbers as they could name, and we came up with: pi, sqrt(2), and e. In fact, these are great examples, but they are the only ones we could name... and we could name infinitely many rational numbers: 1, 2, 3, 4, 5,... and on and on!

Here's a link to a page about irrational numbers.

Rational and Irrational Numbers Worksheet

The purpose of this worksheet is to explore the properties of nonterminating decimals and their fractional equivalents.

Nonterminating Decimals Worksheet

This worksheet explores nonterminating decimals. We didn't do this worksheet in class, but you may find the questions and solutions to be helpful in studying for the first test.

© 2011 Jason B. Hill. All Rights Reserved.