Math 1 · Unit 10 · Lesson 4 · part 3 of 4
Decay: the pennies, and why real data wobbles
Now decay.
Decay: the pennies, and why real data wobbles
Now decay. Put 300 pennies in a cup, shake, and pour them out. Remove every penny that landed heads up. Count what is left, put those back in the cup, and repeat.
| pour | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| pennies left | 300 | 164 | 100 | 46 | 20 | 8 |
Ask what the story suggests first. About half the pennies land heads up each time, so about half are left. The model should be close to
y = 300 · (12)x
a starting value of 300 and a decay factor of 12. Now see what the data says.
The tool gives about y = 348 · 0.49x. The decay factor 0.49 is very close to the story's 12. The starting value came out as 348, not 300.
Why not exactly 300 and one half?Real coin flips are random. On one pour a few more than half land heads; on another, a few fewer. The ratios were 0.55, 0.61, 0.46, 0.43, 0.40, all near one half but none exactly. Real data wobbles, so the best-fitting curve is near the story's model, not on it. Every model is a little bit wrong, and here the story tells you what it was aiming for.
Try fitting the pennies by hand with the Unit 7 explorer. Slide a and b until the curve runs through the dots, then compare your numbers with the tool's.