Simple Chance Experiments
Repeat fair coin flips, fair six-sided die rolls and counter draws. Record every result and compare the counts with predictions based on the chances. Put each counter back and mix before the next draw.
What a learner can do afterwards
- Predict about 10 heads in 20 fair coin flips, tally heads and tails, and compare the recorded counts with the prediction
- Roll a fair six-sided die 30 times, record every result, and compare each number's count with the expected count of about 5
- Draw same-size, same-shape counters from a mixed bag without looking or choosing by feel, record and replace each counter, and compare the results with a prediction
1 · Read
You know words like likely and unlikely. A chance experiment repeats an action with an uncertain result. Choose how many trials to make. Record each result once with a tally mark. The fifth tally crosses the first four. Add the counts to check that every trial was recorded. On a bar chart, read each bar's name and printed number to find its count.
When a coin lands flat, record heads or tails. A fair coin gives both sides an equal chance, so expect about half the flips to be heads and half tails. Find an expected count by dividing the number of flips by 2. Find a missing tails count by subtracting heads from the total. A prediction says what you expect. A recorded count says what happened.
A fair six-sided die has numbers 1 to 6, each with the same chance on every roll. Make a tally column for each number. Divide the total rolls by 6 to find each expected count. For 36 rolls, 36 divided by 6 is 6, so expect about 6 of each. Recorded counts may be higher or lower. Later rolls do not have to balance earlier counts. One count matching its prediction does not prove the die is fair.
For a bag experiment, use counters of the same size and shape. Mix before each draw. Take one without looking or choosing by feel. Record its colour, put it back, and mix again. Putting it back keeps the bag's contents unchanged. With equal numbers of red and blue counters, each colour has an even chance. Expect about half the draws to be red. In 18 draws, half is 9.
With equal chances for red and blue, expect half the draws to be red. In 18 draws, Ada expects 9 red but records 11 red and 7 blue. Subtract expected from recorded: 11 - 9 = 2 more red than expected. To compare the recorded colours, subtract the smaller count from the larger. To combine batches, add each colour's counts across batches.
A fair coin can give a short streak, such as all heads. A streak alone does not prove fairness or unfairness. More trials give more evidence. In larger fair experiments, the shares of outcomes usually get closer to their chances. They need not match exactly, and each extra trial need not bring them closer.
Compare shares when experiments have different totals. One group records 24 heads in 40 flips. Another records 48 heads in 80 flips. Both the total and the heads count doubled, so the heads share is the same. The larger experiment is not closer to half in this example.
A chance experiment lets you test what you expect against what really happens.
Lesson, questions and flashcards written for this dataset.
Readings, practice questions and flashcards written for this dataset. Linked Khan Academy video: 'Making a prediction based on results from a single experiment'.
More to read
Plan and record a coin experiment
Choose how many flips to make before you start. For a fair coin, heads and tails have equal chances. Predict about half of your chosen total for each side. This prediction is different from a recorded result.
Make two tally columns, Heads and Tails. Each time the coin lands flat, add one mark to the matching column. Cross each fifth tally over the previous four. Keep every result, including an unexpected streak.
When you finish, count both columns and add them. Their total should equal the number of recorded flips. If the total does not match, check the record rather than changing counts to fit your prediction. Compare the heads count with half the total. A difference can occur by chance; an exact split is not required.
Compare die counts with a prediction
For a fair six-sided die, each number from 1 to 6 has the same chance. Make six tally columns and record one result for every roll. Before rolling, divide the planned total by 6 to predict each number's count.
For example, 36 rolls give an expected count of about 6 for each number. A recorded count may be above or below 6. Add all six recorded counts to check the total number of rolls.
A number appearing exactly as often as expected does not prove that the die is fair. A number appearing more often than expected does not by itself prove unfairness either. Keep the full record and collect more rolls if you want more evidence. There is no fixed extra number of rolls that guarantees a verdict.
Keep the bag experiment unchanged
Use counters of the same size and shape so you cannot choose a colour by feeling for a different counter. Count the colours before starting and make a prediction. Equal numbers of red and blue counters give each colour an even chance when every counter has the same chance of being drawn.
Mix the bag, draw without looking or selecting by feel, and record the colour. Put the counter back and mix before drawing again. Returning it keeps the colour counts unchanged. Without replacement, the bag's contents could change from one draw to the next.
Afterward, compare your recorded counts with the prediction. Keep unexpected results in the record. To combine batches, add the red counts together and the blue counts together. To compare the two final counts, subtract the smaller from the larger.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Optional connections
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.