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Probabilities of Single Step Events

Probabilities of Multi-step Events

Sampling

Using Samples

This week your student will be working with probability. A probability is a number that represents how likely something is to happen. For example, think about flipping a coin.

Sometimes we can figure out an exact probability. For example, if we pick a random date, the chance that it is on a weekend is 27, because 2 out of every 7 days fall on the weekend. Other times, we can estimate a probability based on what we have observed in the past.

Here is a task to try with your student:

People at a fishing contest are writing down the type of each fish they catch. Here are their results:

Solution:

To find an exact probability, it is important to know what outcomes are possible. For example, to show all the possible outcomes for flipping a coin and rolling a number cube, we can draw this tree diagram:

The branches on this tree diagram represent the 12 possible outcomes, from “heads 1” to “tails 6.” To find the probability of getting heads on the coin and an even number on the number cube, we can see that there are 3 ways this could happen (“heads 2”, “heads 4”, or “heads 6”) out of 12 possible outcomes. That means the probability is 312, or 0.25.

A board game uses cards that say “forward” or “backward” and a spinner numbered from 1 to 5.

This week your student will be working with data. Sometimes we want to know information about a group, but the group is too large for us to be able to ask everyone. It can be useful to collect data from a sample (some of the group) of the population (the whole group). It is important for the sample to resemble the population.

A sample that is selected at random is more likely to be representative of the population than a sample that was selected some other way.

A city council needs to know how many buildings in the city have lead paint, but they don’t have enough time to test all 100,000 buildings in the city. They want to test a sample of buildings that will be representative of the population.

We can use statistics from a sample (a part of the entire group) to estimate information about a population (the entire group). If the sample has more variability (is very spread out), we may not trust the estimate as much as we would if the numbers were closer together. For example, it would be easier to estimate the average height of all 3-year olds than all 40-year olds, because there is a wider range of adult heights.

We can also use samples to help predict whether there is a meaningful difference between two populations, or whether there is a lot of overlap in the data.

Students from seventh grade and ninth grade were selected at random to answer the question, “How many pencils do you have with you right now?” Here are the results:

how many pencils each seventh grade student had

how many pencils each ninth grade student had

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.