Probability Theory

Text before the first section.

Set Theory

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

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

    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)