Chapter 1: Basic Concepts in Probability

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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

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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.

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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.

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1.1 Sample Spaces, Events, and Probabilities

• Definition 1

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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?)

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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?

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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).

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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!

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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.

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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.

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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.
  

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1.2 Simulations

• Simulation Number

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1.2 Simulations

• Simulation Number

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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.

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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.

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1.2 Simulations

• Homework / Class Exercises (Section 1.6, page
  18-20)
• Do problems 1, 8
• Enjoy your homework, and have a
  wonderful day!

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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?

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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?

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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,

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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.

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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,

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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!

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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)

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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)

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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!