Lesson 6: Mathematics: Probability, Statistics and Analytic Geometry
Probability and statistics reward systematic counting, and analytic geometry rewards translating between algebra and pictures. Both areas are full of marks for students who work methodically.
This lesson covers counting, basic probability, mean/median/standard deviation, and the geometry of lines and circles.
Counting and probability
There are 36 equally likely outcomes (6 × 6).
Pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — six pairs.
Probability = 6/36 = 1/6.
C(5,2) = 5!/(2! × 3!) = (5 × 4)/2 = 10.
Probability of at least one head in two tosses: 1 − P(no heads) = 1 − 1/4 = 3/4. The complement trick (‘at least one’ = 1 − ‘none’) saves time.
Mean, median and standard deviation
Mean = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9)/8 = 40/8 = 5.
Median (middle of 8 values, average of 4th and 5th) = (4 + 5)/2 = 4.5.
Variance = [(2−5)² + 3(4−5)² + 2(5−5)² + (7−5)² + (9−5)²]/8 = (9 + 3 + 0 + 4 + 16)/8 = 32/8 = 4.
Standard deviation = √4 = 2.
Lines and circles
Complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4 = 16.
So (x − 3)² + (y + 2)² = 16. Centre (3, −2), radius 4.
Distance = |3(0) + 4(0) − 10| / √(3² + 4²) = 10/5 = 2.
Common pitfalls
Do not count ordered pairs as unordered (or vice versa) in probability: dice pairs (1,6) and (6,1) are distinct outcomes.
Median of an even-sized data set is the average of the two middle values, not one of them.
Further study
- Standard senior-secondary mathematics textbooks (probability, statistics, analytic geometry).
📝 Quiz
1. What is the probability that two dice sum to 7?
Show answer
a) 6 favourable pairs out of 36 outcomes: 6/36 = 1/6.
2. What is C(5,2)?
Show answer
a) (5 × 4)/2 = 10.
3. What is the centre of the circle x² + y² − 6x + 4y − 3 = 0?
Show answer
a) Completing the square gives (x − 3)² + (y + 2)² = 16.