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Unit 5 · 2.1

Sample spaces & probability

n(E) ÷ n(S), dice/cards/coins, complements, "of those who…" percents

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Rule cards

Sample spaces

List every possible outcome: coin + die = 2 × 6 = 12 (H1…T6); two dice = 36 pairs; three dice = 6³ = 216.

Traps
  • Treated (1,2) and (2,1) as one outcome.
    two dice → 21 outcomes
    ✓ the dice are different → 36 ordered pairs
  • Used the wrong number of outcomes.
    three dice → 36
    ✓ three dice → 216

Classical probability

P(E) = n(E) ÷ n(S), reduced; 0 ≤ P ≤ 1; impossible = 0, certain = 1.

  1. Count the outcomes in the event.
  2. Divide by all possible outcomes.
  3. Reduce. Empirical P = times it happened ÷ times tried.
Traps
  • Left the probability unreduced.
    16/52
    ✓ 16/52 = 4/13
  • Counted outcomes in A OR B when the question said AND.
    6 and a spade → 16/52
    ✓ 6 AND spade = the 6♠ only = 1/52

Complement and odds

P(not E) = 1 − P(E); P(at least one) = 1 − P(none); odds against = P(not E) : P(E).

P(not E) (P(Ē) or P(Eᶜ))

Traps
  • Computed P(E) when P(not E) was asked.
    not the 4♦ → 1/52
    ✓ 1 − 1/52 = 51/52
  • Wrote odds in favor when against was asked.
    4 of 100 tickets → 4 : 96
    ✓ against = 96 : 4 = 24 : 1

"Of those who…": the group is the denominator

'Percent OF a group' / 'given' → divide by that group's total only; plain 'AND' → divide by the grand total.

  1. Find the phrase: 'of those…', 'given that…', 'what percent of the … ' — that group is the WHOLE.
  2. Denominator = that group's row or column total.
  3. Numerator = the cell inside that group that matches the rest of the question.
  4. No group named ('a person is male AND …') → denominator = grand total.
  5. Exam Q16/17: P(read) = 1100/1520 = 72%; P(man AND no) = 251/1520 = 17%.
Traps
  • Divided by everyone surveyed when the question named a group.
    % of High-School-degree holders who attended → 400 ÷ 2002 = 20%
    ✓ 400 ÷ 1280 (HS total) = 31.25% ≈ 31% (the key prints 20% — see Key errors)
  • Divided by the wrong margin (reversed the question).
    400 ÷ 684 = 58.5% (that is 'of those who attended')
    ✓ the group after 'of those' is the denominator
  • Used an OR count for an AND question.
    female AND light beer → female total + light total
    ✓ AND = the single cell where the row and column cross

Videos & reading

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