Probability Experiments Students Can Run in One Period

Quick answer: Set the trial count before students start, pool results on the board, and have the extension task printed before the lesson. Two dice and 60 trials per pair takes about fifteen minutes, leaves time for pooling and comparison with theory, and produces a class total near 1,800 trials, which is enough for the pattern to appear clearly.

Why does one group always finish in four minutes?

Because "roll the dice and record what happens" has no defined end. Some pairs roll carefully and tally accurately. Others roll fast, tally approximately, and declare themselves done.

Three fixes. Give an exact number of trials, written on the board, not a time limit. Give a tally sheet with the right number of boxes so finishing is visible. And make the recording sheet require both partners: one rolls and calls, the other tallies, and they swap at the halfway point. A sheet with two named columns enforces this without you saying anything.

Then have the extension ready in paper form on the front desk. Not a harder version of the same thing, and not an early finish. Something that uses their data: predict what 600 trials would give, then compare with the pooled class total when it appears.

Which four experiments fit in one period?

Sum of two dice, 60 trials. The workhorse. There are 36 equally likely outcomes for two distinguishable dice. Six of them give a sum of 7, so the probability is 6 out of 36, or 1 in 6. Only one gives 2 and one gives 12, so those are 1 in 36 each. Students who have drawn the six-by-six grid can predict the shape before they roll, which makes the experiment a test rather than a discovery of something they were told.

Coin runs, 100 flips. Students flip 100 times and record the longest run of the same face. Most classes find runs of 5, 6 or 7. Then ask half the class to fake 100 flips without flipping. Faked sequences almost never contain a run of 6, because people think that looks non-random. You can pick out the fakes from the front of the room and the class finds this unnerving in a productive way.

Two-spinner product, 40 trials. Paper clip and pencil on a printed spinner beats buying anything. Two spinners numbered 1 to 4, multiply the results, record. Sixteen outcomes, and the products are not evenly spread, which surprises students who expect uniformity from uniform parts.

Drawing with and without replacement, 30 trials each. Same bag, two protocols, run back to back. This is the experiment that produces the best arguments, because the difference is small at 30 trials and only becomes convincing when the class pools.

How do you pool the data without losing ten minutes?

Draw the table on the board before the lesson starts, with one row per pair and one column per outcome. Pairs write their own numbers in as they finish. By the time the last pair sits down the table is nearly complete and you add the column totals live.

Then the comparison. Put the theoretical fraction next to each experimental one. For sums of two dice with 15 pairs at 60 trials, the class total is 900 rolls, and the count of sevens should land somewhere near 150. It will not be exactly 150, and the gap is worth two minutes of discussion on its own.

The point to land: individual pairs are all over the place, the class total is close, and the reason is the number of trials. That is the law of large numbers, arrived at from their own tallies rather than from a statement in a textbook.

How do you differentiate?

Approaching: one die, 30 trials, six outcomes. Provide the tally sheet with outcomes already listed and a sentence frame for the conclusion. The skill being assessed is accurate recording and comparing two fractions.

On level: two dice, 60 trials, students construct the 36-outcome grid themselves before rolling and write predicted against observed.

Above level: ask for the expected value of the sum, which is 7, and then the expected value of the product of two dice, which is 12.25. Or move to conditional questions: given that the sum is even, what is the probability it is 8. There are 18 even sums out of 36, and 5 of the 36 outcomes give 8, so the answer is 5 out of 18. That style of question is the whole substance of Conditional Probability and the Four-Field Table.

What goes wrong?

Dice on the floor. Roll inside a shoebox lid or on a folded towel. This is not fussiness, it is the difference between a lesson and forty minutes of retrieving dice.

Miscounting. Insist on tally marks in groups of five and a total written in a box at the end. Students who write running numbers lose their place and quietly invent the last few.

The conclusion that theory is wrong. Some pair will get nine sevens in 60 rolls and announce that the math does not work. Take it seriously, put their number on the board next to the pooled total, and let the pooling answer them.

Frequently asked questions

How many trials are really needed?

For a visible pattern in dice sums, about 500 pooled trials. Individual pairs at 60 will not see it, and telling them that in advance prevents disappointment.

Can I use a random number generator instead?

For the extension, yes, and it lets you reach 10,000 trials in seconds. For the first experience, physical dice are better because students trust their own hands.

What if I have no dice?

Two sets of ten cards numbered 1 to 6, drawn with replacement, behave identically and cost nothing to make.

How do I assess this?

Grade the comparison, not the data. A student whose experimental result is far from theory but who explains the trial count correctly has done the work.

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