diff --git a/publication/publication.ptx b/publication/publication.ptx index 92d6e51..20d3549 100644 --- a/publication/publication.ptx +++ b/publication/publication.ptx @@ -121,11 +121,11 @@ - + /> diff --git a/source/ch-Probability.ptx b/source/ch-Probability.ptx index 7d682af..4eea216 100644 --- a/source/ch-Probability.ptx +++ b/source/ch-Probability.ptx @@ -1,411 +1,547 @@ - Probability Theory + Probability Theory - + +

+ Text before the first section. +

+ + +
+ Set Theory + + +

- Text before the first section. + The sample space, often denoted \Omega, is the set of all possible results of an experiment. + A single result is called an outcome, while a collection of results is called an event.

+ + + + + +

+ An experiment consists of rolling a standard 6-sided die. + The sample space is \Omega = \{1, 2, 3, 4, 5, 6\}. + One possible event is A = \{2, 4, 6\}, i.e., the event that the result of the roll is even. +

+ +

+ What would the sample space look like if we roll the die two times and recorded the results? +

+ + + +

+ + \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), + \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), + \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), + \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), + \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), + \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} + +

+ + + + + +

+ Let A be a set. + The symbol \in means "is an element of", as in a \in A. + Given another set B, we say A is a subset of B, written A\subset B, to mean that every element of the set A is also an element of the set B. +

+ + + + + +

+ Consider sets A and B, each contained inside \Omega. + We can combine sets in a variety of ways: +

+
+
  • + Union + +

    + The union of A and B is the set A \cup B = \{x \mid x \in A \text{ or } x \in B\}. +

    +
    • + +
  • + Intersection + +

    + The intersection of A and B is the set A \cap B = \{x \mid x \in A \text{ and } x \in B\}. +

    +
    • + +
  • + Difference + +

    + The set difference A-B is the set A - B = \{x \mid x \in A \text{ and } x \notin B\}. +

    +
    • + +
  • + Complement + +

    + The complement of A is the set A^c = \{x \in \Omega \mid x \notin A\}. +

    +
    • + +
  • + Empty Set + +

    + The empty set, usually written \emptyset or \{\}, is the set which contains no elements. +

    +
    • +
    +     +     + +

    + It's useful sometimes to draw pictures representing sets... +

    + + + +

    + Two sets A and B are disjoint if A \cap B = \emptyset. +

    +     +     + + + +

    + Given a finite set A, the cardinality of A, written |A|, is the number of elements in A. +

    +     +     + + + + +

    + Consider the sets A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, B = \{2, 4, 9, 10, 12, 14, 19\}, and C = \{9, 10, 11, 14, 16, 17, 20\}, which are all subsets of \Omega = \{1, 2, 3, \dotsc, 20\}. +

    +     + + + + +

    + Find A - (B \cap C). +

    +     + + +

    + \{1, 2, 3, 4, 5, 6, 7, 8\} +

    +     +     + + + + +

    + Find |A|, |B|, |C|, |A\cup B|, |A \cap B|, |B\cap C|, |A\cap C|, and |A\cup B\cup C|. + Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets? +

    +     + + +

    + |A| = 10, |B| = 7, |C| = 7, |A \cup B| = 13, |A \cap B| = 4, |B\cap C| = 3, |A\cap C| = 2, |A\cup B\cup C| = 17. In particular, note that |A\cup B| = 13 \neq 10 + 7 = |A| + |B|, so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets. +

    +     +     + + + + +

    + Find A^c and (A\cup B)^c. +

    +     + + +

    + A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}, (A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}. +

    +     +     +     + + + +

    + Suppose we roll a 6-sided die two times. + List the set of all possible results. + [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result (4, 2)---rolling a 4 and then a 2.] +

    +     +     + + + +

    + Suppose we flip a coin two times. + List the set of all possible results. + What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be? +

    +     +     + + + +

    + If we roll a 6-sided die ten times, how many possible results will there be? +

    +     +     +     +
    + +
    + Definition of Probability + + + +

    + A probability distribution on a sample space \Omega assigns probabilities to every event, satisfying the following conditions: +

    + +
      +
    1. +

      + \Pr(\Omega) = 1. +

      +
      2. + +
    2. +

      + 0 \leq \Pr(A) \leq 1 for any event A. +

      +
      3. + +
    3. +

      + If A \cap B = \emptyset, then \Pr(A\cup B) = \Pr(A) + \Pr(B). +

      +
      4. +
    +     +     + + + + +

    + Consider the sample space \Omega = \{1, 2, 3, 4, 5, 6, 7, 8\} with probability distribution below. + Calculate the probabilities of A = \{1, 3, 7, 8\}, B = \{2, 3, 6, 7\}, A\cup B, and A \cap B. +

    + + + + + + + x + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + + + + \Pr(x) + 0.1 + 0.05 + 0.2 + 0.15 + 0.15 + 0.1 + 0.05 + 0.1 + 0.1 + + +
    + +     +     + + + +

    + Suppose we flip a coin two times. + Answer the questions below. + What about three flips? What about four flips? +

    +     + + + + +

    + Write all outcomes in the sample space \Omega. +

    +     +     + + + + +

    + Make a probability distribution table for \Omega assuming the coin is fair. +

    +     +     + + + + +

    + Make a probability distribution table assuming the coin comes up heads with probability 0.3. +

    +     +     +     + + + +

    + Suppose we roll a die two times. + Answer the questions below. +

    +     + + + + +

    + Write all outcomes in the sample space \Omega. +

    +     +     + + + + +

    + Make a probability distribution table for \Omega assuming the die is fair. +

    +     +     + + + + +

    + Let A be the event that the second roll is higher than the first, and let B be the event that the first roll is even. + Find \Pr(A), \Pr(B), and \Pr(A \mid B). +

    +     +     +     + + + +

    + Suppose a die has the values 1, 2, 3, 4, 5, 6 on the faces, but the die is not fair. + Instead, the probabilities scale by the same amount as the face values. + For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. + Write a probability distribution table for this die. +

    +     +     + + + +

    + Suppose a die has the values 1, 2, 3, 4, 5, 6 on the faces, but the die is not fair. + Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. + Write a probability distribution table for this die. +

    +     +     + + + +

    + A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. + For each value of n = 1, 2, 3, \dotsc, find the probability of the toxin molecule leaving the cell during the nth minute. + What is the probability of the molecule leaving the cell during the first 3 minutes? +

    +     +     + + + +

    + Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. + For each value of n = 1, 2, 3, \dotsc, and for each value of 0\leq k \leq n, find the probability that exactly k toxin molecules remain in the cell after the nth minute. +

    +     +     +     +
    + +
    + Conditional Probability + +

    + Text of section. +

    + + + +

    + In each of the following scenarios with given events A and B, alculate \Pr(A), \Pr(B), \Pr(A\cap B), \Pr(A \mid B), and \Pr(B \mid A). +

    -
    - Set Theory - + +

    - Text of section. + An experiment consists of rolling a fair die two times. + Let A be the event that the sum is even, and let B be the event that the second roll is higher than the first. +

    +     +     + + + +

    + An experiment consists of flipping a fair coin three times. + Let A be the event that the first and second flips match. + Let B be the event that there are at least two heads. +

    +     +     + + + + +

    + A diagnostic test is developed to detect a disease present in 3.2% of the population. + For a patient who has the disease, the test will accurately give a positive result 65% of the time. + When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time. +

    +     + + + + +

    + For a patient who receives a positive test, what is the probability they have the disease? +

    +     + + +

    + TODO +

    +     +     + + + + +

    + For a patient who receives a negative test, what is the probability they do not have the disease? +

    +     + + +

    + TODO +

    +     +     +     + +
    + +
    + Independent Events + +

    + Text of section. +

    + + + + +

    + An experiment consists of rolling a fair die two times. + Let A be the event that the sum is even, and let B be the event that the second roll is higher than the first. + Are A and B independent? +

    +     +     + + + +

    + An experiment consists of flipping a fair coin three times. + Let A be the event that the first and second flips match. + Let B be the event that there are at least two heads. + Are A and B independent? +

    +     +     + + + +

    + Let A = \{1, 2, 3\} and B = \{3, 4, 5\}. + Fill in the following probability distribution table so that A, B are independent.

    - - - -

    - Consider the sets A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, B = \{2, 4, 9, 10, 12, 14, 19\}, and C = \{9, 10, 11, 14, 16, 17, 20\}, which are all subsets of \Omega = \{1, 2, 3, \dotsc, 20\}. -

    -     + + + + + x + 1 + 2 + 3 + 4 + 5 + 6 + - - -

    - Find A - (B \cap C). -

    -     - - -

    - \{1, 2, 3, 4, 5, 6, 7, 8\} -

    -     -     - - - - -

    - Find |A|, |B|, |C|, |A\cup B|, |A \cap B|, |B\cap C|, |A\cap C|, and |A\cup B\cup C|. - Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets? -

    -     - - -

    - |A| = 10, |B| = 7, |C| = 7, |A \cup B| = 13, |A \cap B| = 4, |B\cap C| = 3, |A\cap C| = 2, |A\cup B\cup C| = 17. In particular, note that |A\cup B| = 13 \neq 10 + 7 = |A| + |B|, so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets. -

    -     -     - - - - -

    - Find A^c and (A\cup B)^c. -

    -     - - -

    - A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}, (A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}. -

    -     -     - - - - -

    - Suppose we roll a 6-sided die two times. - List the set of all possible results. - [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result (4, 2)---rolling a 4 and then a 2.] -

    -     -     - - - -

    - Suppose we flip a coin two times. - List the set of all possible results. - What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be? -

    -     -     - - - -

    - If we roll a 6-sided die ten times, how many possible results will there be? -

    -     -     - - - -
    - Definition of Probability - -

    - Text of section. -

    - - - - -

    - Consider the sample space \Omega = \{1, 2, 3, 4, 5, 6, 7, 8\} with probability distribution below. - Calculate the probabilities of A = \{1, 3, 7, 8\}, B = \{2, 3, 6, 7\}, A\cup B, and A \cap B. -

    - -
    - - - - - x - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - - - - \Pr(x) - 0.1 - 0.05 - 0.2 - 0.15 - 0.15 - 0.1 - 0.05 - 0.1 - 0.1 - - -
    - -     -     - - - -

    - Suppose we flip a coin two times. - Answer the questions below. - What about three flips? What about four flips? -

    -     - - - - -

    - Write all outcomes in the sample space \Omega. -

    -     -     - - - - -

    - Make a probability distribution table for \Omega assuming the coin is fair. -

    -     -     - - - - -

    - Make a probability distribution table assuming the coin comes up heads with probability 0.3. -

    -     -     -     - - - -

    - Suppose we roll a die two times. - Answer the questions below. -

    -     - - - - -

    - Write all outcomes in the sample space \Omega. -

    -     -     - - - - -

    - Make a probability distribution table for \Omega assuming the die is fair. -

    -     -     - - - - -

    - Let A be the event that the second roll is higher than the first, and let B be the event that the first roll is even. - Find \Pr(A), \Pr(B), and \Pr(A \mid B). -

    -     -     -     - - - -

    - Suppose a die has the values 1, 2, 3, 4, 5, 6 on the faces, but the die is not fair. - Instead, the probabilities scale by the same amount as the face values. - For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. - Write a probability distribution table for this die. -

    -     -     - - - -

    - Suppose a die has the values 1, 2, 3, 4, 5, 6 on the faces, but the die is not fair. - Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. - Write a probability distribution table for this die. -

    -     -     - - - -

    - A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. - For each value of n = 1, 2, 3, \dotsc, find the probability of the toxin molecule leaving the cell during the nth minute. - What is the probability of the molecule leaving the cell during the first 3 minutes? -

    -     -     - - - -

    - Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. - For each value of n = 1, 2, 3, \dotsc, and for each value of 0\leq k \leq n, find the probability that exactly k toxin molecules remain in the cell after the nth minute. -

    -     -     -     -
    - -
    - Conditional Probability - -

    - Text of section. -

    - - - -

    - In each of the following scenarios with given events A and B, alculate \Pr(A), \Pr(B), \Pr(A\cap B), \Pr(A \mid B), and \Pr(B \mid A). -

    -     - - - -

    - An experiment consists of rolling a fair die two times. - Let A be the event that the sum is even, and let B be the event that the second roll is higher than the first. -

    -     -     - - - -

    - An experiment consists of flipping a fair coin three times. - Let A be the event that the first and second flips match. - Let B be the event that there are at least two heads. -

    -     -     -     - - - -

    - A diagnostic test is developed to detect a disease present in 3.2% of the population. - For a patient who has the disease, the test will accurately give a positive result 65% of the time. - When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time. -

    -     - - - - -

    - For a patient who receives a positive test, what is the probability they have the disease? -

    -     - - -

    - TODO -

    -     -     - - - - -

    - For a patient who receives a negative test, what is the probability they do not have the disease? -

    -     - - -

    - TODO -

    -     -     -     -     -
    - -
    - Independent Events - -

    - Text of section. -

    - - - - -

    - An experiment consists of rolling a fair die two times. - Let A be the event that the sum is even, and let B be the event that the second roll is higher than the first. - Are A and B independent? -

    -     -     - - - -

    - An experiment consists of flipping a fair coin three times. - Let A be the event that the first and second flips match. - Let B be the event that there are at least two heads. - Are A and B independent? -

    -     -     - - - -

    - Let A = \{1, 2, 3\} and B = \{3, 4, 5\}. - Fill in the following probability distribution table so that A, B are independent. -

    - - - - - - - x - 1 - 2 - 3 - 4 - 5 - 6 - - - - \Pr(x) - - - - - - - - -
    -     -     -     -
    + + \Pr(x) + + + + + + + + + + + +     +
    \ No newline at end of file diff --git a/source/frontmatter.ptx b/source/frontmatter.ptx index eab9ebd..34e27c2 100644 --- a/source/frontmatter.ptx +++ b/source/frontmatter.ptx @@ -7,21 +7,21 @@ - You - Your department - Your institution + Andy Eisenberg + Mathematics Department + Temple University - + - 20202024 - You + 2025 + Andy Eisenberg This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit CreativeCommons.org