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Chapter 1 Basic Concepts in Probability

- It will include the following topics/sections
- 1.1 Sample Spaces, Events, and Probabilities
- 1.2 Simulations
- 1.3 Complementary Events and Mutually Exclusive

Events - 1.4 Some Probability Rules

1.1 Sample Spaces, Events, and Probabilities

- A sample space, S, is defined as the set of all

possible outcomes of an experiment. If the

outcomes in S are equally likely, we call S an

equally probable sample space. Any subset of S is

called an event.

1.1 Sample Spaces, Events, and Probabilities

- Example 1If a box contains three balls numbered

1, 2, 3 and we draw one ball from the box in such

a way that each of the three balls are equally

likely to be selected, then 1, 2, 3 is an

equally probable sample space and - For this experiment odd, even is also a

sample space but not an equally probable sample

space since an odd outcome is twice as likely as

an even outcome.

1.1 Sample Spaces, Events, and Probabilities

- Definition 1

1.1 Sample Spaces, Events, and Probabilities

- Example 2 Box I contains balls numbered 1, 2, 3

and box II contains 2, 3 and 4. One ball is

selected at random from each box. Then, the

sample space is S(1, 2), (1, 3), (1, 4), (2,

2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4),

where the first number in each order pair denotes

the number on the ball drawn from box I and the

second number in each ordered pair denotes the

number on the ball drawn from box II. This is the

set of all possible outcomes of this experiment,

and S is an equally probable sample space.

Consider the event A where Asum of digits on

the selected balls is an even number. Then

A(1, 3),(2, 2),(2, 4),(3, 3). Hence P(A)4/9.

(Why?)

1.1 Sample Spaces, Events, and Probabilities

- Example 3 Two red blocks and two green blocks

are arranged at (all the possible arrangements

are equally likely) in a row. Let Athere is

exactly one green block between two red blocks.

Find P(A). - Hint S(RRGG),(RGRG),(RGGR),(GGRR),(GRGR),(GRRG

), and S is equally probable sample space.

Why?

1.1 Sample Spaces, Events, and Probabilities

- Example 4 A box contains four balls numbered 1,

2, 3, 4. Two chips are drawn, at random, without

replacement, from the box. Let Alargest number

selected is 3. Find P(A).

1.1 Sample Spaces, Events, and Probabilities

- Homework / Class Exercises (Section 1.6, page 19)
- Do problems 2, 3, 6
- Enjoy your homework, and have a

wonderful day!

1.2 Simulations

- There are situations where either there is

no an analytic solution or we dont know to find

the analytic solution to a probability problem.

In these cases we can estimate P(A) by long-run

proportion of the times that A occurs. In other

words, we simulate the experiment. For this

course, we will use a random number target (see

page 6) to carry out these simulations.

1.2 Simulations

- Example 1 Box I contains balls numbered 1,

2, 3 and box II contains balls numbered 2, 3, 4.

One ball is selected at random from each box.

Consider the event Athe sum of the digits on

the selected balls is an even number. To

demonstrate the simulation procedure we will

estimate P(A) in this example.

1.2 Simulations

- Let us simulate this experiment 20 times

(why?). For the drawing from box I, we select

a number from the random number target until we

get either a 1, 2, or 3. So for this drawing,

4s, 5s, and 6s will be ignored. For the

drawing from box II, we select a number from the

random number target until we get either a 2, 3,

or 4. In this case 1, 5, and 6 are ignored.

1.2 Simulations

- Simulation Number

1.2 Simulations

- Simulation Number

1.2 Simulations

- The estimate of P(A) is the proportion of

times which A occurs which is 8/20 or 0.4. This

is not P(A). We have already evaluated P(A) to be

4/9. The value of 0.4 is an estimate of P(A)

based on the simulations.

1.2 Simulations

- Example 2 Problem 9, section 1.6
- Describe how to estimate P(A) for the above

example without ignoring any numbers from the

random number target.

1.2 Simulations

- Homework / Class Exercises (Section 1.6, page

18-20) - Do problems 1, 8
- Enjoy your homework, and have a

wonderful day!

1.3 Complementary Events and Mutually Exclusive

Events

- Definition
- Events A, B are said to be mutually exclusive if

they cannot occur together. - Example 1 Toss a die once. Then the events A1

and 5 are mutually exclusive. Why?

1.3 Complementary Events and Mutually Exclusive

Events

- Definition
- Events A, B are said to be complementary if both

of the following conditions are satisfied - Events A, B are mutually exclusive.
- Events A, B exhaust the sample space in the sense

that every outcome in S is ether in A or in B. - Example 2 Toss a die once. Then the events

Aodd number and Beven number are complementary.

Why?

1.3 Complementary Events and Mutually Exclusive

Events

- Example 3 Suppose that student Meat Loaf takes 5

courses for each of the 8 semester that he is

working on his college degree. Describe the

complementary event to each of the following - Gat least one semester Meat Loaf gets all As,
- HMeat Loaf makes at least one A every

semester,

1.3 Complementary Events and Mutually Exclusive

Events

- Solution (i) This means that Meat Loaf gets 5

As in either 1, 2, 3, 4, 5, 6, 7, or 8

semesters. The complementary event is that he

makes 5 As zero times. So, not Ghe makes less

than 5 As every semester. - (ii) This means that Meat Loaf makes either 1, 2,

3, 4, or 5 As for all eight semesters. The

complementary event is that he makes zero As in

at least one semester. So, not Hhe makes zero

As in at least one semester.

1.3 Complementary Events and Mutually Exclusive

Events

- Exercise Suppose that student Meat Loaf takes 2

courses for each of the 2 semester that he is

working on his college degree. - (a) Describe the sample space.
- (b) Describe the complementary event to each of

the following both in terms of English statement,

and mathematical notation. - Gat least one semester Meat Loaf gets all As,
- HMeat Loaf makes at least one A every

semester,

1.3 Complementary Events and Mutually Exclusive

Events

- Homework / Class Exercises (Section 1.6, page

18-20) - Do problems 10
- Enjoy your homework, and have a

wonderful day!

1.4 Some Probability Rules

- Probabilities satisfy the following rules
- Rule 1 P(A)0 if A is impossible event (Athe

empty set) - Rule 2 P(A)1 of A is certain event (AS)
- Rule 3 P(A)1-P(not A)
- Rule 4 If A and B are mutually exclusive events,

P(A or B)P(A)P(B)

1.4 Some Probability Rules

- Example 1Look Example 1.8 on page 10.
- Choose one letter from English language text.
- Find
- P(vowel is chosen)
- P(consonant is chosen)

1.4 Some Probability Rules

- Homework / Class Exercises (Section 1.6, page

18-20) - Do problems 11, 15
- Enjoy your homework, and have a

wonderful day!

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