Math 1070Q - 4.3 – Sample Spaces and Events Notes

Recap Section 4.2

  • Cardinality of a set: The number of elements in a set A is denoted by n(A).
  • Rules for union of sets:
    • The cardinality of an empty set is n(\emptyset) = 0.
    • For two sets A and B: n(A \cup B) = n(A) + n(B) - n(A \cap B).
    • For three sets A, B, and C: 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: These are useful graphical tools for visualizing relationships between three or more sets.

Basic Probability Terminology

  • Experiment: An activity that produces observable results. The observed results must be definite and not vague (e.g., measurements, weights).
    • Example (1): Flipping a coin
      • Experiment: "tossing / flipping a coin".
      • Outcomes: head (H) or tail (T). Landing on an edge is considered rare and not a definite outcome.
    • Example (2): Rolling a die
      • Experiment: "rolling a die".
      • Outcomes: 1, 2, 3, 4, 5, or 6 dots.
  • Outcome: A single result of an experiment.
  • Sample Space (S): The set of all possible outcomes of an experiment. It usually denoted by S.
    • Example (1) Continuation: For flipping a coin, S = {H, T}.
    • Example (2) Continuation: For rolling a die, S = {1, 2, 3, 4, 5, 6}.
  • Trial: Each repetition of an experiment.
  • Determining Sample Spaces for Multi-step Experiments:
    • When an experiment involves multiple steps, the outcomes can be ordered pairs or tuples.
    • Example: Flipping a coin twice
      • The outcomes of the second flip are independent of the first.
      • Order matters: (head, tail) is different from (tail, head).
      • Notation: (a, b) where 'a' is the result of the 1st flip and 'b' is the result of the 2nd flip.
      • Sample Space: S = {(H, H), (H, T), (T, H), (T, T)}.

Tree Diagrams

  • Purpose: To systematically list all possible outcomes of a multi-step experiment, especially when there are many possibilities, preventing outcomes from being missed.
  • Method: Start from a root point and draw branches for each possible outcome at each step of the experiment.
  • Example (3): Rolling a die, then flipping a coin if even, then flipping again if tail
    • Step 1: Roll a die. Six initial branches: 1, 2, 3, 4, 5, 6.
    • Step 2: If the die shows an even number (2, 4, or 6), flip a coin. From branches 2, 4, and 6, draw two sub-branches each for H and T.
    • Step 3: If the coin shows tail (from Step 2), flip it again. From each 'T' sub-branch (originating from 2, 4, 6), draw two more sub-branches for H and T.
    • Determining Outcomes: Trace along each complete path from the root to the end of a branch. Each path represents an outcome as an ordered tuple of all elements along that path.
    • Sample Space (S):
      • Outcomes from odd die rolls: 1, 3, 5 (these branches end after the first roll).
      • Outcomes from even die rolls followed by H: (2, H), (4, H), (6, H).
      • Outcomes from even die rolls followed by T, then H: (2, T, H), (4, T, H), (6, T, H).
      • Outcomes from even die rolls followed by T, then T: (2, T, T), (4, T, T), (6, T, T).
      • Full 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)}. There are 12 branches/outcomes.

Events

  • Definition: Given a sample space S for an experiment, an event E is any subset of S.
  • Simple Event: An event with only a single outcome, meaning it is a subset of S containing exactly one element.
    • Example (4): Rolling a die
      • Event E: "an even number of dots shows up" is E = {2, 4, 6}.
    • Example (4) Continuation: Tossing a coin
      • Event E: "the coin shows tail" is a simple event, E = {T}.

Relation with Set Theory

Probability terminology often renames concepts from set theory:

  • Set theory term $\rightarrow$ Probability term
    • Universal set $\rightarrow$ Sample space or Certainty event
    • Empty set $\rightarrow$ Impossible event
    • Intersection of two sets $\rightarrow$ Intersection of two events
    • Union of two sets $\rightarrow$ Union of two events
    • Complement of a set $\rightarrow$ Complement of an event
    • Two disjoint sets $\rightarrow$ Two mutually exclusive events
  • Examples to illustrate set theory connections:
    • Example (5): Flipping a coin
      • The event "the coin lands in head AND tail" is an impossible event because H and T are mutually exclusive outcomes for a single flip.
    • Example (6): Rolling a die
      • Event E: "the die shows an even number of dots" (E = {2, 4, 6}).
      • Event F: "the die shows 3" (F = {3}).
      • Since 3 is not an even number, E \cap F = \emptyset. Therefore, E and F are mutually exclusive events.

Worked Example: Rolling a Die Twice

  • Example (7): Consider the experiment of 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."

  • Solution Steps:

    1. Find the Sample Space (S):
      • Rolling a die once yields 6 outcomes. Rolling it twice means 6 \times 6 = 36 total outcomes.
      • Each outcome is an ordered pair (x, y) where x is the first roll and y is the second roll.
      • S = {(1, 1), (1, 2), …, (6, 6)} (listing all 36 pairs).
    2. List all elements of E and F:
      • Event E: First roll is an even number (2, 4, or 6). The second roll can be anything (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)}.
        • n(E) = 18.
      • Event F: The sum of dots from both rolls is at least 10 (i.e., 10, 11, or 12, since the maximum sum is 6+6=12).
        • Sum = 10: (4, 6), (6, 4), (5, 5) (3 possibilities).
        • Sum = 11: (5, 6), (6, 5) (2 possibilities).
        • Sum = 12: (6, 6) (1 possibility).
        • F = {(4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)}.
        • n(F) = 6.
    3. List all elements of the event E $\cap$ F. Are E and F mutually exclusive?
      • E $\cap$ F: Contains outcomes that are in both E (first roll is even) and F (sum is at least 10).
        • From the roster of F, check which ones have an even first roll: (4, 6), (6, 4), (6, 5), (6, 6).
        • E \cap F = {(4, 6), (6, 4), (6, 5), (6, 6)}.
        • Since E \cap F is not empty (n(E \cap F) = 4), events E and F are not mutually exclusive.
    4. Find the complement of the event E $\cup$ F.
      • E $\cup$ F: The union of all outcomes in E or F.
        • E \cup F = {(2, 1), …, (2, 6), (4, 1), …, (4, 5), (4, 6), (5, 5), (5, 6), (6, 1), …, (6, 6)}.
        • More systematically, using n(E \cup F) = n(E) + n(F) - n(E \cap F) = 18 + 6 - 4 = 20.
        • So, 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)}.
      • $(\text{E} \cup \text{F})^c$: The complement of E $\cup$ F includes all elements in S that are not in E $\cup$ F.
        • n((\text{E} \cup \text{F})^c) = n(S) - n(E \cup F) = 36 - 20 = 16.
        • (\text{E} \cup \text{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

  • Concept: Some sample spaces and events can be infinite, often described by a range of values rather than a list of distinct outcomes.
    • These are given using set-builder notation.
  • Example (8): Measuring water volume in a tank
    • Experiment: Measuring the volume of water in a tank with a 200-gallon capacity.
    • Sample Space (S): The volume v can be any real number between 0 and 200 gallons.
      • S = {v \mid 0 \le v \le 200, v \text{ in gallons}}.
    • Event E: "the content of the tank is strictly between 50 and 150 gallons" (meaning, not including 50 or 150).
      • E = {x \mid 50 < x < 150, x \text{ in gallons}}.

More Information: The French Deck of Cards

  • Standard Deck: A deck consists of 52 cards.
  • Suits: Four suits:
    • Black: Spade (\clubsuit) and Club (\spadesuit).
    • Red: Heart (\heartsuit) and Diamond (\diamondsuit).
  • Cards per Suit: Each suit has 13 cards:
    • Numbered cards: 2, 3, 4, 5, 6, 7, 8, 9, 10.
    • Special cards: Jack (J), Queen (Q), King (K), Ace (A).
  • Numerical Values: J=11, Q=12, K=13, A can be 1 or 14 depending on the game.
  • Example (9): Drawing a card from the deck
    • Experiment: Drawing a single card from a 52-card deck.
    • Sample Space (S): The set of all 52 unique cards.
    • Event E: "A card is either a spade or a 2."
      • This includes all 13 spades AND all the 2s that are not already spades.
      • 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 of spades is listed only once, as it satisfies both conditions.)
      • n(E) = 13 \text{ (spades)} + 3 \text{ (other 2s)} = 16.