Math 1070Q: Sample Spaces and Events

Math 1070Q - Mathematics for Business and Economics: 4.3 – Sample Spaces and Events

Recap Section 4.2

  • Cardinality of a set: n(A) = \text{number of elements of A}.
  • Rules for union of two sets and three sets:
    • n(\emptyset) = 0
    • n(A \cup B) = n(A) + n(B) - n(A \cap B)
    • n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)
  • Venn diagrams are useful for three or more sets.

This Lecture

  • Basic probability terminology.
  • Tree diagrams.
  • Relation between probability and set theory.
  • Introduction to continuous sample spaces.

Terminology: Experiment and Outcomes

  • Experiment: An activity that yields observable results. 'Observable' means definite and not vague (e.g., a measurement, weight).
  • Outcome: A specific result of an experiment.
  • Example 1: Flipping a coin.
    • Experiment: "tossing / flipping a coin".
    • Outcomes: head (H), tail (T). Landing on its edge is rare and not considered a definite outcome.
  • Example 2: Rolling a die.
    • Experiment: "rolling a die".
    • Outcomes: 1, 2, 3, 4, 5, 6 dots facing upward.

Terminology: Sample Spaces and Trials

  • Sample Space (S): The set of all possible outcomes of an experiment.
  • Trial: Each repetition of an experiment.
  • Example 1: Flipping a coin.
    • Sample space S = {H, T}.
  • Example 2: Rolling a die.
    • Sample space S = {1, 2, 3, 4, 5, 6}.

Determining Sample Spaces

  • The concept of "chance to win the lottery today is 1 in a million" implies that there are 1 \text{ million} lottery tickets issued, and each has an equal likelihood of winning.
  • For experiments with multiple steps, outcomes can be ordered pairs or tuples.
    • Example: Flipping a coin twice. The second flip is independent of the first.
      • An outcome is an ordered pair (a, b), where a is the result of the 1st flip and b is the result of the 2nd flip.
      • Order matters: (head and tail) \neq (tail and head).
      • Possible outcomes: (H, H), (H, T), (T, H), (T, T).
      • Sample space: S = {(H, H), (H, T), (T, H), (T, T)}.

Tree Diagrams

  • Tree diagrams are useful tools to visualize and keep track of all possible outcomes in multi-step experiments, especially when outcomes might be missed.
  • Example 3: Determine the sample space for the following experiment:
    • Step 1: Roll a die.
    • Step 2: If the die shows an even number (2, 4,, or 6), flip a coin.
    • Step 3: If the coin shows tail (T) in Step 2, flip it again.
  • Solution using a Tree Diagram:
    • Initial Roll (Root to 1st branch): 6 initial outcomes from rolling a die: 1, 2, 3, 4, 5, 6.
    • Coin Flip (2nd branch): From even outcomes (2, 4, 6), draw 2 branches each for H and T.
    • Second Coin Flip (3rd branch): From any 'T' outcome in the previous coin flip, draw 2 more branches for H and T.
    • To find an outcome, trace from the root to the end of a branch. The outcome is the ordered tuple of all elements along that branch. For instance, the sequence 2 \rightarrow T \rightarrow H gives the outcome (2, T, H).
    • There are 12 distinct branches/outcomes.
    • Resulting Sample Space: S = {1, 3, 5, (2, H), (4, H), (6, H), (2, T, H), (2, T, T), (4, T, H), (4, T, T), (6, T, H), (6, T, T)}.

Events

  • Event (E): A subset of the sample space S for an experiment.
  • Simple Event: An event consisting of a single outcome (a subset of S with only one element).
  • Example 4:
    • In rolling a die, event E: "an even number of dots shows up" is E = {2, 4, 6}.
    • In tossing a coin, event E: "the coin shows tail" is a simple event, E = {T}.

Relations with Set Theory

  • Probability terminology often renames concepts from set theory:
    • Set theory terms \leftrightarrow Probability terms
    • Universal set \leftrightarrow Sample space or Certainty event
    • Empty set \leftrightarrow Impossible event
    • Intersection of two sets \leftrightarrow Intersection of two events
    • Union of two sets \leftrightarrow Union of two events
    • Complement of a set \leftrightarrow Complement of an event
    • Two disjoint sets \leftrightarrow Two mutually exclusive events
  • Example 5: In flipping a coin, the event "the coin lands in head AND tail" is an impossible event (equivalent to the empty set \emptyset ).
  • Example 6: In rolling a die, let E: "the die shows an even number of dots" (E = {2, 4, 6}) and F: "the die shows 3" (F = {3}).
    • Since 3 is not even, E and F have no common outcomes (E \cap F = \emptyset).
    • Therefore, E and F are mutually exclusive events.

Worked Example: Rolling a Die Twice

  • Experiment: Rolling a die twice.
  • Events Defined:
    • E: "the number of dots in the first roll is even."
    • F: "the sum of dots from both rolls is at least 10.
  • Part 1: Find the Sample Space (S)
    • Each roll has 6 outcomes. Rolling twice gives 6 \times 6 = 36 total outcomes.
    • S = {(1, 1), (1, 2), …, (6, 6)} (all 36 ordered pairs).
  • Part 2: List all elements of E and F
    • Event E: First roll is even (2, 4, \text{ or } 6); second roll can be any (1-6).
      • E = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),\ (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),\ (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} (18 outcomes).
    • Event F: Sum of dots is at least 10 (i.e., 10, 11,, or 12).
      • Sum = 10: (4, 6), (6, 4), (5, 5).
      • Sum = 11: (5, 6), (6, 5).
      • Sum = 12: (6, 6).
      • F = {(4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)} (6 outcomes).
  • Part 3: List all elements of the event E \cap F. Are E and F mutually exclusive?
    • E \cap F = {(4, 6), (6, 4), (6, 5), (6, 6)}.
    • Since E \cap F is not empty, E and F are not mutually exclusive.
  • Part 4: Find the complement of the event E \cup F
    • First, find E \cup F by combining elements of E and F, removing duplicates:
      • E \cup F = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),\ (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),\ (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6),\ (5, 5), (5, 6)}. (There are 18 + 6 - 4 = 20 elements).
    • The complement (E \cup F)^c includes all elements in S that are not in E \cup F.
      • Since n(S) = 36 and n(E \cup F) = 20, then n((E \cup F)^c) = 36 - 20 = 16.
      • (E \cup F)^c = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),\ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),\ (5, 1), (5, 2), (5, 3), (5, 4)}.

Continuous Sample Spaces

  • Some sample spaces and events can be infinite, often described using set-builder notation rather than rosters.
  • Example 8: A water tank has a capacity of 200 gallons.
    • Experiment: Measuring the volume of water in the tank.
    • Sample Space: S = {v \mid 0 \leq v \leq 200, v \text{ in gallons}}. This represents all real numbers between 0 and 200, inclusive.
    • Event E: "the content of the tank is strictly between 50 and 150 gallons."
    • E = {x \mid 50 < x < 150, x \text{ in gallons}}. "Strictly" means the endpoints (50 and 150) are not included.

More Information: The French Deck of Cards

  • A standard deck has 52 cards.
  • Four Suits: Spade (♠), Club (♣) are black; Heart (♡), Diamond (♢) are red.
  • Cards per Suit: 13 cards per suit.
    • Numbered cards: 2 to 10 (9 cards).
    • Special cards: Jack (J), Queen (Q), King (K), Ace (A) (4 cards).
    • Numerical values: J = 11, Q = 12, K = 13. Ace (A) can be 1 or 14 depending on the game.
  • Example 9: Drawing a card from the deck.
    • Sample Space (S): The set of all 52 cards.
    • Event E: "A card is either a spade or a 2."
    • E = {2\spadesuit, 2\heartsuit, 2\clubsuit, 2\diamondsuit, 3\spadesuit, 4\spadesuit, 5\spadesuit, 6\spadesuit, 7\spadesuit, 8\spadesuit, 9\spadesuit, 10\spadesuit, J\spadesuit, Q\spadesuit, K\spadesuit, A\spadesuit}. (Note: The 2\spadesuit card is included only once, as it satisfies both conditions.)