Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Some probabilities cannot be worked out by counting equally likely outcomes. What is the chance that a dropped thumbtack lands point up, or that a paper cup tossed in the air lands on its side? The honest way to answer is to run the experiment many times and keep score. The fraction of trials in which the event happened, its relative frequency, becomes an estimate of the probability.
Seventh graders discover that relative frequency wobbles a lot over a handful of trials and settles down as the number of trials grows. Ten flips of a coin might give 7 heads; a thousand flips will almost always give something much closer to half. That settling is why larger experiments are trusted more than small ones.
The idea also runs the other way. If a probability is known, students multiply it by the number of trials to predict roughly how often the event will occur. With a fair number cube, rolling a 3 or a 6 has probability 2/6 = 1/3, so in 600 rolls they expect about 200, while understanding that the real count will probably be a little above or below that.
Expecting exactly 200 threes and sixes in 600 rolls misreads a prediction. Chance causes variation, so 193 or 211 are perfectly normal results.
A class that flips a coin 10 times and gets 8 heads may decide the coin is biased. Combining results from the whole class shows the relative frequency moving toward 1/2.
After several heads in a row, many students expect tails next. Each flip of a fair coin is independent, so the chance of tails is still 1/2.
A class tosses a paper cup 250 times. It lands on its side 160 times, open end up 35 times and open end down 55 times. Estimate the probability of each outcome, then predict how many times it would land on its side in 1,000 tosses.
Answer: Estimated probabilities: side 0.64, open end up 0.14, open end down 0.22. In 1,000 tosses expect about 640 side landings, though not exactly 640.
Pool data from the whole class into a running total and plot relative frequency against the number of trials. The graph zigzags early on and flattens later, which is far more convincing than any explanation. Free random-number tools let students extend to thousands of trials in seconds.
Assessment typically asks for a relative frequency from a table, an estimate of how many times an event will occur, or a judgment about whether a set of results suggests the outcomes are fair. Ask students to use the words 'about' or 'approximately' whenever they make a prediction.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 0.35 (also accepted: .35, 7/20, 35%)
Relative frequency = 28 ÷ 80 = 0.35.
Answer: 80
P(blue) = 1/5, and 1/5 × 400 = 80. The actual count will be close to 80 but probably not exactly 80.
Answer: B) Flipping it 500 times
Relative frequency settles down as the number of trials grows, so 500 flips give the most trustworthy estimate.
Answer: C) 1/2
The coin has no memory. Each flip is independent, so the probability of heads stays 1/2.
Answer: 255
Multiply the probability by the number of trials: 0.85 × 300 = 255 seeds, approximately.
Answer: A) The result is close to the expected 200, so it fits a fair cube
The expected count is 1/3 × 600 = 200. A result of 214 is a normal amount of variation, so it does not suggest unfairness.
A full lesson with slides, activities and an exit ticket on relative frequency and long-run probability, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.SP.C.6 with an answer key, ready in about a minute.
Make a worksheet →Turn relative frequency and long-run probability into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Not quite. Relative frequency is what happened in an experiment; probability is the long-run value it gets close to. With many trials the two are usually close.
There is no fixed number, but more is better. In class, pooling everyone's results to reach a few hundred trials usually makes the pattern clear.