Deeper

Lab · c11.s02.L1

Urn and dice

The size of a sample space is counted by construction: each step multiplies the choices, and describing an outcome more coarsely changes the count.

Question: How many outcomes are there?

Urn and diceAn urn of six numbered balls, beside two six-sided dice. A choice tree grows one column per draw: six branches, then six again, each leaf one ordered outcome, so the counter reads 6 × 6 = 36. Keeping the ball out shrinks the second level to five: 6 × 5 = 30, and dividing out the two orders gives 15. A third draw gives 216, or 120 without replacement. The dice count 36 ordered outcomes but only 11 sums.Urn and diceSample spaces and countingOBJECTCOUNT1234566 balls6 = 66outcomesEvery step offers the samenumber of choices.draw 1start1234566 OUTCOMES — one cell per tuplegold: ball 1123456
Schematic Schematic: the urn and the dice are drawings, not measured objects, and every count is an exact enumeration computed by data/scripts/c11.s02.L1.py. The frames multiply the choices at each step — 6 × 6 = 36 with replacement, 6 × 5 = 30 without it, 6 × 5 ÷ 2 = 15 when order is ignored, and 216 or 120 at a third step — then show two dice that agree at 36 while collapsing to 11 distinct sums.
Lab controls
Urn

Left or Up selects Urn, Right or Down selects Dice; Home selects Urn, End selects Dice; Space or Enter confirms

6 choices

Left or Down one step, Right or Up one step, Page Down or Page Up two steps, Home 2, End 6

2 draws

Left or Down one step, Right or Up one step, Home 1, End 3

1100 ms per frame

Left or Down one step, Right or Up one step, Page Down or Page Up five steps, Home 300, End 1600

Frame 1 of 9: Six balls in the urnThe urn holds six numbered balls. One draw offers six choices, so the sample space has six outcomes: 6 = 6.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

An urn of six numbered balls, beside two six-sided dice. A choice tree grows one column per draw: six branches, then six again, each leaf one ordered outcome, so the counter reads 6 × 6 = 36. Keeping the ball out shrinks the second level to five: 6 × 5 = 30, and dividing out the two orders gives 15. A third draw gives 216, or 120 without replacement. The dice count 36 ordered outcomes but only 11 sums.

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Caption status: schematic. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.