Probability and Statistics: Counting, Chance and Averages

Formulas (2)

• The multiplication principle and arrangements (permutations) · Arrangements
  $${}^nP_r = \frac{n!}{(n - r)!} \qquad \text{with repeats: } \frac{n!}{p!\,q!\cdots}$$
• Combinations: choosing a group when order does not matter · Combinations
  $$\binom{n}{r} = \frac{n!}{r!\,(n - r)!}$$

Watch out for (2)

• Identical letters do not make new arrangements
  → The multiplication principle and arrangements (permutations)
• Ask whether order matters before you count
  → Combinations: choosing a group when order does not matter

Formulas (4)

• Probability from equally likely outcomes, and the complement · Equally likely outcomes
  $$P(A) = \frac{n(A)}{n(S)} \qquad P(\text{not } A) = 1 - P(A)$$
• The addition rule for 'A or B' · Addition rule
  $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
• Independent events and tree diagrams (with replacement) · Independent events
  $$P(A \cap B) = P(A)\,P(B)$$
• Conditional probability and drawing without replacement · Conditional probability
  $$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$$

Watch out for (5)

• With two dice, (2, 5) and (5, 2) are different outcomes
  → Probability from equally likely outcomes, and the complement
• Subtract the overlap when events can happen together
  → The addition rule for 'A or B'
• Exactly one needs every order
  → Independent events and tree diagrams (with replacement)
• Without replacement, the second fraction changes
  → Conditional probability and drawing without replacement
• P(B given A) is not P(A given B)
  → Conditional probability and drawing without replacement

Formulas (3)

• Mean, median, mode and range · Mean
  $$\bar{x} = \frac{\sum x}{n} \qquad \sum x = n\,\bar{x}$$
• Combined and weighted means, and how averages change when data change · Combined mean
  $$\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$$
• Frequency tables and standard deviation (in outline) · Mean of a frequency table and standard deviation
  $$\bar{x} = \frac{\sum f x}{\sum f} \qquad \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$$

Watch out for (3)

• Order the data before finding the median
  → Mean, median, mode and range
• A combined mean is not the average of the two means
  → Combined and weighted means, and how averages change when data change
• Shifting every value does not change the spread
  → Frequency tables and standard deviation (in outline)