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Home›Probability and statistics

Probability

A fair coin lands heads half the time, but only in the long run: John Kerrich flipped one 10,000 times and got 5,067 heads. Flip thousands of coins, roll dice, drop balls through a Galton board and change the sample size, then check every result against the exact probability.

Progress

Before you start

Probabilities here are written as fractions, decimals and sometimes powers, such as 2/6 or 1 − (5/6)⁴, so Fractions helps if those feel rusty. The last unit uses a mean and a standard deviation, which Averages introduces. Everything else is built from coins, dice and a Galton board in the page.

When you finish

You will be able to list outcomes for a die or two coins, explain why Kerrich's 5,067 heads in 10,000 flips fits a fair coin, and show why 7 is the likeliest sum of two dice. You can find the chance of at least one six in four rolls, read conditional probability from a table, count Galton board paths, and compute a standard error.

  1. 1Outcomes, events and probabilityA fair die has six faces, so the chance of rolling at least a five is 2 out of 6, about 0.333.
  2. 2The law of large numbersJohn Kerrich flipped a coin 10,000 times during the Second World War and got 5,067 heads, a proportion of 0.5067.
  3. 3Two dice and the sum of sevenTwo dice can land in 36 equally likely ways, and 6 of them add up to 7, more than any other sum.
  4. 4NOT, OR and ANDThe chance of at least one six in four rolls of a die is 1 − (5/6)⁴ = 671/1296, about 0.518.
  5. 5Conditional probability and independenceIn an OpenStax sample of 100 people, 13 are left-handed, but among the 48 females only 4 are, about 0.083.
  6. 6The Galton board and the bell curveOn a Galton board with 10 rows of pegs, a ball lands in the middle bin with probability 252/1024, about 0.246.
  7. 7Sample size and the central limit theoremThe mean of 25 values from a population with standard deviation 15 has a standard error of 15 ÷ √25 = 3.
Sources for this course (9)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 5~6학년군 [6수04-04]~[6수04-06] 가능성을 말과 0, 1/2, 1 같은 수로 나타내기; 중학교 1~3학년군 [9수04-03] 상대도수, [9수04-05] 경우의 수(두 경우의 수를 더하거나 곱하는 정도), [9수04-06] 확률(상대도수로서의 의미와 경우의 수의 비율로서의 의미, 각 경우의 가능성이 같다는 가정), 용어 상대도수·사건·확률; 공통수학1 [10공수1-03-01]~[10공수1-03-03] 합의 법칙·곱의 법칙·순열·조합, 기호 nCr; 확률과 통계 [12확통01-03] 이항정리, 용어 이항계수·파스칼의 삼각형; [12확통02-01]~[12확통02-06] 확률의 기본 성질·덧셈정리·여사건의 확률·조건부확률·독립과 종속·곱셈정리, 용어 시행·통계적 확률·수학적 확률·여사건·배반사건·조건부확률·종속·독립·독립시행과 기호 P(A), P(B|A); [12확통03-03] 이항분포, [12확통03-04] 정규분포와 이항분포의 관계, [12확통03-05]·[12확통03-06] 임의추출·표본평균과 모평균, 용어 큰 수의 법칙·이항분포·정규분포·모평균·표본평균과 기호 B(n, p), N(m, σ²), X̄)
  • Textbook: OpenStax, Introductory Statistics 2e by Barbara Illowsky and Susan Dean, 3.1 terminology, equally likely outcomes, dime and nickel, law of large numbers, Belgian one euro coin 56% heads in 250 trials, conditional probability, Table 3.1 handedness of 100 people (openstax.org/books/introductory-statistics-2e/pages/3-1-terminology)
  • 4.3 binomial distribution
  • 7.1 central limit theorem for sample means, example mean 90, SD 15, n = 25, standard error 3 (.../pages/7-1-the-central-limit-theorem-for-sample-means-averages)
  • Yut rules for the Korean example only: 국립민속박물관 한국민속대백과사전 「윷놀이」, 도·개·걸·윷·모 by how many of the four sticks land 뒤집어진 (folkency.nfm.go.kr/kr/topic/detail/1595, read 2026-10-19)
  • Kerrich story: John Kerrich, 10,000 tosses, 5,067 heads (JMP case study JMP014 by DeWayne Derryberry, jmp.com/en/academic/case-study-library/is-a-coin-fair, citing Baldi and Moore; Wikipedia "John Edmund Kerrich")
  • Galileo story: Sopra le scoperte dei dadi, sum 10 in 27 ways and 9 in 25 ways with three dice (English translation, University of York history of statistics, york.ac.uk/depts/maths/histstat/galileo.pdf)
  • Pascal story: Pascal to Fermat, 29 July 1654, one six in four throws "as 671 to 625" (American Physical Society, aps.org/apsnews/2009/07/pascal-letters-fermat-points; Gaston Sanchez, De Mere's games; Statistical Science 2024 "Antoine Gombaud, Chevalier de Méré")
  • Galton story: Francis Galton, Natural Inheritance of 1889, the quincunx (Wikipedia "Galton board")

Checked: 2026-10-17