Probability: SPI Mathematics

Formulas (3)

• Experiment, outcome, sample space and event · Number of outcomes
  $$n(S) = \text{number of outcomes in the sample space}$$
• Probability as favourable outcomes over all outcomes · Classical probability
  $$P(E) = \frac{n(E)}{n(S)}$$
• Listing the outcomes for two or three coins · Outcomes for k coins or dice
  $$n(S) = 2^k \text{ for } k \text{ coins} \qquad n(S) = 6^k \text{ for } k \text{ dice}$$

Reference tables (1)

Watch out for (6)

• Forgetting to simplify
  → Probability as favourable outcomes over all outcomes
• 'Not even' is not a new count
  → Probability as favourable outcomes over all outcomes
• Counting 1 as a prime
  → Which faces of a die are prime, composite or square
• Counting 2 as an odd prime
  → Which faces of a die are prime, composite or square
• Treating HT and TH as one outcome
  → Listing the outcomes for two or three coins
• 'At most one' includes none
  → Listing the outcomes for two or three coins

Formulas (2)

• Every probability lies from 0 to 1 · Range of a probability
  $$0 \le P(E) \le 1$$
• The probability of an event not happening · Complement rule
  $$P(\text{not } E) = 1 - P(E)$$

Watch out for (4)

• Reading a percentage as a whole number
  → Every probability lies from 0 to 1
• Allowing negative probabilities
  → Every probability lies from 0 to 1
• Subtracting from 100 in the wrong column
  → The probability of an event not happening
• Answering with the event instead of its complement
  → The probability of an event not happening

Formulas (3)

• The 36 outcomes of two dice · Ways to make a sum s with two dice
  $$\text{ways}(s) = 6 - |s - 7|, \quad 2 \le s \le 12$$
• Both dice at once: multiply, or use the complement · Two independent dice
  $$P(A \text{ and } B) = P(A) \times P(B) \qquad P(\text{at least one}) = 1 - P(\text{none})$$
• Pairs of days: 53 Sundays and two visitors · Two people choosing from d days
  $$P(\text{same day}) = \frac{d}{d^2} = \frac{1}{d} \qquad P(\text{different days}) = 1 - \frac{1}{d}$$

Watch out for (6)

• Counting (2, 5) and (5, 2) once
  → The 36 outcomes of two dice
• Counting a sum twice in 'divisible by 4 or 6'
  → The 36 outcomes of two dice
• 'Between' leaves out the end values
  → The 36 outcomes of two dice
• Adding instead of using the complement
  → Both dice at once: multiply, or use the complement
• An even product needs only one even number
  → Both dice at once: multiply, or use the complement
• Answering 1/7 for a leap year
  → Pairs of days: 53 Sundays and two visitors

Formulas (3)

• Bags of balls, numbered cards and letters of a word · Drawing one item
  $$P = \frac{\text{items that fit}}{\text{all items}} \qquad \text{numbers } a \text{ to } b: \; b - a + 1 \text{ cards}$$
• Drawing two cards or balls together · Two drawn together
  $$P(\text{both of a kind}) = \frac{a}{n} \times \frac{a-1}{n-1}$$
• Odds in favour and odds against · Odds and probability
  $$\text{odds in favour} = f : u, \quad P(E) = \frac{f}{f + u}$$

Reference tables (1)

Watch out for (7)

• All the kings, or only the black ones
  → The make-up of a 52-card deck
• Counting the ace as a face card
  → The make-up of a 52-card deck
• Cards from a to b are b minus a plus 1
  → Bags of balls, numbered cards and letters of a word
• Counting a repeated letter once
  → Bags of balls, numbered cards and letters of a word
• Keeping 52 in the second fraction
  → Drawing two cards or balls together
• Odds are not a probability
  → Odds in favour and odds against
• Counting the ace of spades twice
  → Odds in favour and odds against

Formulas (2)

• Experimental probability from a record of trials · Experimental probability
  $$P(E) = \frac{\text{times } E \text{ happened}}{\text{total trials}}$$
• Expected number from a probability · Expected count
  $$\text{expected number} = P(E) \times \text{total}$$

Watch out for (3)

• Answering 1/2 for a recorded coin
  → Experimental probability from a record of trials
• Giving the count of the event, not the one asked
  → Expected number from a probability
• Dropping a decimal place
  → Expected number from a probability