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Sections 1.1 and 1.2 lecture notes Math331, Fall 2008 Instructor: David Anderson
Section 1.
Non-mathematical Definition: In any experiment, an event that may or may not happen is called random. Ex: weather, outcome of biological experiment, dice, cards...
How to make sense of this notion? And how to calculate?
Good idea: Relative frequency interpretation. To find the “probability” that an event A occurs in an experiment. Let n(A) be the number of times that A occurs during n performances of the experiment. Finally, define
probability(A) = p(A) = lim n→∞
n(A) n
Example 1. Rolling a fair die. Will find p(6 is rolled) ≈ 1 /6.
Example 2. Flipping a fair coin twice. Will find p(first heads, then tails) ≈ 1 /4.
Problems:
- Can not be computed exactly.
- No reason to believe that limn→∞ n(A)/n always exists.
- Notions that do not have repeatability do not have meaning (examples: weather, guilt/innocence in criminal cases).
More rigor is needed. In fact, we need set theory for an understanding of probability.
Section 1.2: Sample spaces and events.
Definition 1. The sample space of an experiment, S, is the set of all possible outcomes of that experiment. Each individual outcome is called a sample point or point. Subsets of S are called events.
- Have found: confusion of problems in this class typically come from not understanding the sample space.
Example 3. A coin is tossed twice and the outcome of each is recorded. Then, S = {HH, HT, T H, T T }. The event that the second toss was a Head is {HH, T H}.
Example 4. Consider 3 lightbulbs. Our experiment consists of finding out which lightbulb burns out first, and how long (in hours) it takes for this to happen.
S = {(i, t) : i ∈ { 1 , 2 , 3 }, t ≥ 0 },
i tells you which one burns out, and t gives how long it lasted, in hours. The event that the 2nd bulb burns out first, and it lasts less than 3 hours is the set {(2, t) : t < 3 } ⊂ S.
Example 5. A bus arrives at a bus stop sometime between 11 PM and 11:30 PM every night. Give a sample space for the arrival time of the bus.
S = {t : t ∈ [0, 30]}, where t is number of minutes past 11.
or S = {t : t ∈ [11 : 00, 11 : 30]} where t is the time.
Example 6. You roll a four sided die until a 4 comes up. The event you are interested in is getting a three with the first two rolls.
S = {a 1 a 2 · · · an | n ≥ 1 , an = 4, ai ∈ { 1 , 2 , 3 } for i 6 = n} E = { 33 a 3 a 4 · · · an | n ≥ 3 , an = 4, ai ∈ { 1 , 2 , 3 } for i 6 = n}
Definition 2. E is a subset of F if x ∈ E implies x ∈ F. Notation: E ⊂ F.
Definition 3. E and F are equal, denoted E = F , if E ⊂ F and F ⊂ E.
Definition 4. The intersection of E and F , denoted EF or E ∩ F , is
{x ∈ S : x ∈ E and x ∈ F }.
Definition 5. The union of E and F , denoted E ∪ F , is
{x ∈ S : x ∈ E or x ∈ F }.
Definition 6. The complement of E, denoted Ec, is {x ∈ S : x /∈ E}. ∅ = Sc.
Definition 7. The difference of E and F is E − F = {x ∈ S : x ∈ E and x /∈ F }.
Definition 8. Two sets are mutually exclusive if E ∩ F = ∅.
The sets
⋃n i=1 Ei,^
i=1 Ei,^
⋂n i=1 Ei,^
i=1 Ei^ are defined in obvious way.
Discuss Venn Diagrams - no rigor. (Ec^ ∩ G) ∪ F is good ex.
Important set relations.
Commutative law: E ∪ F = F ∪ E, E ∩ F = F ∩ E.
Associative law:
E ∪ (F ∪ G) = (E ∪ F ) ∪ G, E ∩ (F ∩ G) = (E ∩ F ) ∩ G.
