Common Core Standards: Math
6. 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.
It isn't too much of a leap to go from the very basic, informal understanding of probability in 7.SP.5 to get to the formal definition of probability: the number of favorable outcomes divided by the number of total outcomes possible.
Once students know how to calculate probabilities, put 'em to work with coins, dice, spinners, and basically anything they can use to generate frequencies and calculate probabilities. Once they've generated enough data, they can use it to approximate the probabilities of these events. (Disclaimer: what's "enough data" is totes your call. Whether you want them to flip that coin 50 or 500 times is up to you. Just remember, you were once in their shoes, too. Have mercy.)
After calculating probabilities, they should be able to make use of those probabilities to calculate the frequency of a "favorable outcome" if the event were to be repeated a given number of times. Students have been around the block, though, and they should understand that calculating relative frequencies from probabilities provide estimations, not final, conclusive numbers.
If students are having trouble grasping this, have them flip a fair coin two times. They'll see that even though they know the probability of tossing a heads to be , they won't always get 2 heads. Some students will get 2 tails and others will get 2 heads—and that's okay! Because ultimately, no matter what we do and no matter how many times we repeat an event, the results are still subject to the gods of chance.
This would also be a good opportunity to introduce your students to the ideas of theoretical and experimental probabilities, and the differences between them. (Actually, "relative frequency" is Common Core code for "experimental probability.") Plus, this standard discusses the "long-run relative frequency" of an event, which is subject to the Law of Large Numbers. As we repeat an event more and more, the experimental probability of the event will approach the theoretical probability. In other words, the more trials we perform, the more accurate we'll be.
If students are having trouble grasping this, have them flip a fair coin a hundred times and compare this to their two-flip trial. They'll see that their experimental probability will be closer to the theoretical probability of than before.
Er, well, probably.
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