8.1.1: Mystery Bags
Lesson
Let's make predictions based on what we know.
Exercise \(\PageIndex{1}\): Going Fishing
Andre and his dad have been fishing for 2 hours. In that time, they have caught 9 bluegills and 1 yellow perch.
The next time Andre gets a bite, what kind of fish do you think it will be? Explain your reasoning.
Exercise \(\PageIndex{2}\): Playing the Block Game
Your teacher will give your group a bag of colored blocks.
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Follow these instructions to play one round of the game:
- Everyone in the group records the color written on the bag in the first column of the table.
- Without looking in the bag, one person takes out one of the blocks and shows it to the group.
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If they get a block that is the same color as the bag, they earn:
- 1 point during round 1
- 2 points during round 2
- 3 points during round 3
- Next, they put the block back into the bag, shake the bag to mix up the blocks, and pass the bag to the next person in the group.
- Repeat these steps until everyone in your group has had 4 turns.
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At the end of the round, record each person's score in the table.
What color bag? person 1's score person 2's score person 3's score person 4's score round 1 round 2 round 3 Table \(\PageIndex{1}\) - Pause here so your teacher can give you a new bag of blocks for the next round.
- Repeat the previous steps to play rounds 2 and 3 of the game.
- After you finish playing all 3 rounds, calculate the total score for each person in your group.
Are you ready for more?
Tyler’s class played the block game using purple, orange, and yellow bags of blocks.
- During round 1, Tyler’s group picked 4 purple blocks and 12 blocks of other colors.
- During round 2, Tyler’s group picked 11 orange blocks and 5 blocks of other colors.
- During round 3, Tyler forgot to record how many yellow blocks his group picked.
For a final round, Tyler’s group can pick one block from any of the three bags. Tyler’s group decides that picking from the orange bag would give them the best chance of winning, and that picking from the purple bag would give them the worst chance of winning. What results from the yellow bag could have lead Tyler’s group to this conclusion? Explain your reasoning.
Summary
One of the main ways that humans learn is by repeating experiments and observing the results. Babies learn that dropping their cup makes it hit the floor with a loud noise by repeating this action over and over. Scientists learn about nature by observing the results of repeated experiments again and again. With enough data about the results of experiments, we can begin to predict what may happen if the experiment is repeated in the future. For example, a baseball player who has gotten a hit 33 out of 100 times at bat might be expected to get a hit about 33% of his times at bat in the future as well.
In some cases, we can predict the chances of things happening based on our knowledge of the situation. For example, a coin should land heads up about 50% of the time due to the symmetry of the coin.
In other cases, there are too many unknowns to predict the chances of things happening. For example, the chances of rain tomorrow are based on similar weather conditions we have observed in the past. In these situations, we can experiment, using past results to estimate chances.
Practice
Exercise \(\PageIndex{3}\)
Lin is interested in how many of her classmates watch her favorite TV show, so she starts asking around at lunch. She gets the following responses:
\(\text{yes yes yes no no no no no no no yes no no no}\)
If she asks one more person randomly in the cafeteria, do you think they will say “yes” or “no”? Explain your reasoning.
Exercise \(\PageIndex{4}\)
An engineer tests the strength of a new material by seeing how much weight it can hold before breaking. Previous tests have held these weights in pounds:
\(1,200\quad 1,400\quad 1,300\quad 1,500\quad 950\qquad 1,600\qquad 1,100\)
Do you think that this material will be able to hold more than 1,000 pounds in the next test? Explain your reasoning.
Exercise \(\PageIndex{5}\)
A company tests two new products to make sure they last for more than a year.
- Product 1 had 950 out of 1,000 test items last for more than a year.
- Product 2 had 150 out of 200 last for more than a year.
If you had to choose one of these two products to use for more than a year, which one is more likely to last? Explain your reasoning.
Exercise \(\PageIndex{6}\)
Put these numbers in order from least to greatest.
\(\frac{1}{2}\qquad\frac{1}{3}\qquad\frac{2}{5}\qquad 0.6\qquad 0.3\)
Exercise \(\PageIndex{7}\)
A small staircase is made so that the horizontal piece of each step is 10 inches long and 25 inches wide. Each step is 5 inches above the previous one. What is the surface area of this staircase?
(From Unit 7.3.5)