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.)