C

  • Event space vs. sample space

    • Sample space S: all possible outcomes of six coin flips

    • Event space E: subset of S we care about (e.g., X = number of heads)

    • For fair coins, each outcome has equal probability

    • In six flips, |S| = 2^6 = 64, total outcomes

  • Counting outcomes for six flips

    • Fundamental counting principle: each flip has 2 outcomes; over 6 flips: |S| = 2^6 = 64

    • Number of ways to get exactly k heads: inom{6}{k}

    • Examples: inom{6}{3} = 20, inom{6}{0} = 1, inom{6}{1} = 6

    • Therefore, exactly 3 heads: 20 outcomes; not exactly 3 heads: 64 − 20 = 44 outcomes

  • Exact vs not exactly three heads

    • P(X = 3) = rac{20}{64} = 0.3125

    • P(X ≠ 3) = 1 − P(X = 3) = rac{44}{64} = 0.6875

    • Complements: P(E) + P(E^c) = 1; P(E^c) = 1 − P(E)

  • Probability notation and definitions

    • P(E): probability that event E occurs

    • E can be {X = 3} or {X ≠ 3}; X is a random variable, here "number of heads"

    • If you write P(X = 3), that is the probability that the random variable X equals 3

    • Keep fractions unsimplified when that preserves information (e.g., 20/64 instead of 5/16)

    • Probabilities lie in the interval 0 \le P(E) \le 1

    • Impossible event: P(E) = 0; Certain event: P(E) = 1

  • Random variable and common examples

    • Let X = ext{number of heads in 6 flips}

    • Then the event space examples: ext{E}1 = {X = 3}, ext{E}2 = {X \neq 3}

    • The sample space is the same across events: all 64 outcomes

  • Quick observations and intuition

    • Distribution of heads in 6 flips is symmetric around 3; the most likely exact count is 3 heads

    • The phrasing of probability questions matters (exact vs not, complementary events, etc.)

    • For binomial-type problems with a fair coin, the distribution is binomial with parameters n = 6, p = \tfrac{1}{2}

  • Quick course logistics (open-note quiz context)

    • Quiz is described as open-note and open-neighbor; you can consult with others

    • Access via the Mathematize portal; may require creating an account

    • Cost to access platform mentioned; plan accordingly if needed

    • Deadline noted for the initial quiz window (check current syllabus for updates)

  • Key formulas to remember

    • Number of outcomes: |S| = 2^n for n coin flips

    • Exactly k heads: ext{ways} = inom{n}{k}

    • Probability of exactly k heads: P(X = k) = rac{inom{n}{k}}{2^n} for a fair coin

    • Complement rule: P(E^c) = 1 - P(E) and P(E) + P(E^c) = 1

  • Exact vs not exactly three heads

    • P(X = 3) = \frac{20}{64} = 0.3125

    • P(X \neq 3) = 1 \text{

  • } P(X = 3) = \frac{44}{64} = 0.6875

    • Complements: P(E) + P(E^c) = 1

    • P(E^c) = 1 \text{

  • } P(E)

  • Probability notation and definitions

    • P(E): probability that event E occurs

    • E can be {X = 3} or {X \neq 3}; X is a random variable, here "number of heads"

    • If you write P(X = 3), that is the probability that the random variable X equals 3

    • Keep fractions unsimplified (\text{e.g., } \frac{20}{64} \text{ instead of } \frac{5}{16})

    • Probabilities lie in the interval 0 \le P(E) \le 1

    • Impossible event: P(E) = 0; Certain event: P(E) = 1

  • Random variable and common examples

    • Let X = \text{number of heads in 6 flips}

    • Then the event space examples: \text{E}1 = {X = 3}, \text{E}2 = {X \neq 3}

    • The sample space is the same across events: all 64 outcomes

  • Quick observations and intuition

    • Distribution of heads in 6 flips is symmetric around 3; the most likely exact count is 3 heads

    • The phrasing of probability questions matters (exact vs not, complementary events, etc.)

    • For binomial-type problems with a fair coin, the distribution is binomial with parameters n = 6, p = \frac{1}{2}

  • Quick course logistics (open-note quiz context)

    • Quiz is described as open-note and open-neighbor; you can consult with others

    • Access via the Mathematize portal; may require creating an account

    • Cost to access platform mentioned; plan accordingly if needed

    • Deadline noted for the initial quiz window (check current syllabus for updates)

  • Key formulas to remember

    • Number of outcomes: \left|S\right| = 2^n for n coin flips

    • Exactly k heads: \text{ways} = \binom{n}{k}

    • Probability of exactly k heads: P(X = k) = \frac{\binom{n}{k}}{2^n} for a fair coin

    • Complement rule: P(E^c) = 1 \text{

  • } P(E) and P(E) + P(E^c) = 1