# VectorPaint three-toss distribution: group eight sequences into one-three-three-one probability bars
Language: en
Canonical: [VectorPaint three-toss distribution: group eight sequences into one-three-three-one probability bars]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T10:05:49+08:00
Author: Bison
Declare independent fair trials, group by head count and avoid assigning equal probability to four possible counts.
English translation of the [Chinese original](), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.
Three tosses yield four possible head counts, zero through three, but not four equally likely values. List eight equally likely ordered sequences, group them 1,3,3,1 and draw probability bars in VectorPaint. These are deductions under independent fair trials, not invented records of actual experiments.
## Declare three independent fair trials
State n=3, outcomes H/T each probability 1/2 and independent trials. Define X as total heads, not a toss position, with values 0,1,2,3. Use an approximately 1800×1400 canvas with sequence and distribution panels under one title. Record any real experiment separately, without claiming verified fairness. Text/shape tools arrange the model; independence is not inferred automatically from a coin icon.
## List each of eight ordered outcomes once
List HHH,HHT,HTH,THH,HTT,THT,TTH,TTT, each three ordered symbols. There are 2×2×2=8, each probability (1/2)³=1/8. HHT and HTH remain distinct here despite two heads; HH is incomplete. Check duplicates and omissions, not only total row count. Preserve the same H/T codes with explanations in both languages.
## Group by the number of H symbols
X=0 contains TTT; X=1 contains HTT,THT,TTH; X=2 contains HHT,HTH,THH; X=3 contains HHH. Counts 1,3,3,1 total 8, yielding probabilities 1/8,3/8,3/8,1/8. Label each row’s X before grouping, not by its first symbol. Add mutually exclusive sequences rather than assigning 1/4 to each count. Keep numeric group names and the original ledger after recolouring.
## Draw four probability bars on one scale
Place equally spaced X centres at screen x=400,650,900,1150 and baseline y=1100. At 1600 px per probability unit, heights are 200,600,600,200 and tops y=900,500,500,900; use width 150. Keep one vertical scale. Gaps indicate discrete X, not a smooth continuous density. Bar width is layout, not another probability factor.
## Check total probability and expectation 1.5
The four probabilities sum to 8/8=1. The expectation is 0/8+3/8+6/8+3/8=12/8=1.5. Real X remains integer: 1.5 is neither a required outcome nor a fifth probability bar. Put expectation in a separate note. Heights 1,3,3,1 without division by eight describe counts, not a probability-labelled axis.
## Separate model probabilities from finite frequencies
For later real rounds, record sequence/X before observed counts rather than copying 1,3,3,1. Even eight rounds need not include every sequence once. Changed head probability p or dependence invalidates this equal-weight model and requires recalculation, not merely deleting “fair” from the title. No random generator or automatic simulation is required for this static diagram, and the reference guarantees no exact finite-experiment match.
## Troubleshoot equal-quarter and missing-sequence errors
For four equal bars inspect atom/count confusion. For unequal X=1/2 groups inspect missing/duplicated sequences. If a 600-pixel bar is labelled 1/8, correct its label and bar association rather than move the axes. If HHT is assigned to X=1, count each H instead of guessing from a prefix. After each correction check membership, total 8, probability sum 1 and common scale. Keep codes and text independently editable so font changes do not alter content. The app does not prove model assumptions.
## Reopen and trace sequences into the probability bars
Save native work, export PNG and reopen. Read assumptions/eight sequences and trace HHT to X=2 and THT to X=1, then inspect denominator 8 and probability axis. Enlarge text rather than deleting independence/fairness. Reopen editable sequence/group/bar objects and retain sum/expectation calculations. Explain unequal count probabilities and non-outcome expectation without claiming actual tossing, tested fairness or automatic simulation. Recalculate changed trials, probabilities or dependence.
## Related tutorials and sources
[VectorPaint five-row Pascal triangles: check adjacent sums and row totals](); [VectorPaint two-dice sum maps: distinguish thirty-six outcomes from eleven totals]().
Sources checked on 15 November 2026: the [official VectorPaint App Store listing]() and [Apple public lookup]() show version 1.0.8, requiring iOS / iPadOS 17.0 or later. Draw manually with published shapes, paths, nodes, layers and text. Conditions, scales and layouts are original exercise arrangements; check installed controls. Keep editable native work and PNG previews. SVG/PDF may retain any bitmap content used. External calculations and static diagrams are not new automatic proof, statistics or solver features. No completed device, physical-object or universal export testing is claimed.
Probability: [OpenStax’s binomial-distribution chapter]() specifies fixed trials, two outcomes, independence and constant probability. The eight-sequence exercise/layout is original; observed frequency is not substituted for theoretical probability.