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Unit 7 · 2.3

Multiplication rules & conditional probability

AND → multiply; "given" → that group's total is the denominator

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Rule cards

AND, independent events

Independent (one doesn't change the other: coin & die, with replacement, a fixed %) → P(A and B) = P(A) × P(B).

  1. Coin + die: tail AND even = 1/2 × 3/6 = 1/4 (exam Q15).
  2. Repeated % events: none of 4 text (35% do) = (.65)⁴ = .179.
  3. Both drives fail at 3% each: .03² = .0009.
Traps
  • Added the probabilities for AND.
    head and 4 → 1/2 + 1/6
    ✓ 1/2 × 1/6 = 1/12 (exam Q14)
  • Shrank the pool when the item was replaced.
    two queens replaced → 4/52 × 3/51
    ✓ replaced → (4/52)² = 1/169

AND, without replacement

Without replacement the pool shrinks: P(A and B) = P(A) × P(B given that A has occurred).

P(A and B) = P(A) · P(B given that A has occurred) (P(A ∩ B) = P(A) · P(B | A))

Traps
  • Kept the same denominator after not replacing.
    green then yellow, no replacement → 5/16 × 6/16
    ✓ 5/16 × 6/15 = 1/8
  • Shrank the total but not the count of the same kind.
    two face cards → 12/52 × 12/51
    ✓ 12/52 × 11/51 = 11/221

"AND" vs "GIVEN" in a two-way table

AND → one cell ÷ the grand total. GIVEN A → that cell ÷ A's row or column total only.

P(B given that A has occurred) = P(A and B) ÷ P(A) (P(B | A) = P(A ∩ B) / P(A))

  1. Underline the word after 'given that' (or 'of those who'). That group is your new whole.
  2. Denominator = that row/column TOTAL.
  3. Numerator = the cell in that row/column that matches the other event.
  4. No 'given'? AND → cell ÷ grand total; OR → (row + column − cell) ÷ grand total.
  5. From percents: 25% passed both, 42% passed the first → P(2nd given 1st) = .25 ÷ .42 = 60%.
Traps
  • Swapped which group is given.
    P(read given man) → 534/1100
    ✓ given MAN → men's total: 534/785 = 68% (exam Q21); 534/1100 = 49% is P(man given read)
  • Divided by the grand total for a 'given' question.
    P(read given man) → 534/1520 = 35%
    ✓ given man → only the Men row: 534/785
  • Used the wrong total as the base for a percent.
    of those with a Bachelor's → ÷ everyone
    ✓ the group after 'given' / 'of those' is the whole

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