By Christopher Danielson

In the Common Core State Standards, probability and statistics are intertwined. High school students commit to probability models and then test those models by collecting data. Similarly, they use probability to describe how likely the conclusions they draw from their data are.

Probability is about “How likely?” questions. Statistics is about “What is the relationship?” questions about sets of data. The relationships you identify in answering statistics questions always involve probability. And vice versa — answers to probability questions are based on having collected data.

Here is an example that demonstrates this relationship.

From time to time, soft drink companies run promotions in which you can win a prize if a specific code is printed inside the can. You can’t see the code until you buy and drink the beverage. You know that your can has a chance of winning when you take it from the store shelf, but you don’t know whether it’s a winning can. A typical claim in this kind of promotion is that “one in six wins,” which means that of all the cans in this promotion, one‐sixth of them are winners.

That one‐sixth of the cans are winners is a statistics claim. Each can is measured (in a sense) as either a winner or a loser. The data on all of the cans is summarized — that’s doing statistics. Each can has a one‐sixth chance of being a winner — that’s a probability claim.

Now you decide to buy a six‐pack of this beverage. You may do so believing that you’ll be sure to get a winner. The probability question there is: “How likely are you to get a winner if you buy a six‐pack?” Your sureness about winning when you buy a six‐pack is based on statistics — the data summary that one‐sixth of the cans are winners.

Even if you think you’ll certainly get a winner, you’re answering a probability question; you’re saying to yourself that the answer is 100 percent. Now you buy yourself a six‐pack and are surprised to find no winner in it. This data should force you to change your thinking about the probability.

In the end, the probability of getting at least one winner when you buy a six-pack is about

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High school students can find this result using probability theory.

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Also they can find it using statistics. Rolling a die once can represent one can, which is either a winner (if you roll a 1) or a loser (if you roll anything else). Rolling a die six times represents buying a six-pack. If students roll a large number of groups of six rolls, and if they keep track of the number of six-packs with wins, they’ll end up with something quite close to

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of these six-packs having at least one winner.