Probability and statistics: Dropping things onto a target

Today we decided to teach our littles some probability and statistics.

We tested this on our two children, who are aged 4 and 8. The 4-year-old was able to participate in the experiment, but (not too surprisingly) couldn't do the multiplication required to make the predictions. The 8-year-old could do the math. Both said they wanted to do something like this tomorrow.

For this activity you will need:

1. Something high up for your budding statisticians to stand on and drop things off. We used the top of our stairs.
2. Nine pieces of paper, all the same size
3. A lot of soft balls to drop from the high place onto the pieces of paper. E.g. scrunched up pieces of paper

Setup

1. Mark one of the nine pieces of paper with an O
2. Lay out the nine pieces of paper in a rectangle on the floor below your dropping place, like this:
   
   

   	O

Ours actually ended up looking like this, and we plan to use it for other activities:

The numbers are not important for this activity, but it sure did let the kiddos get excited and yell "ooh, I got a five!"

Taping it to some cardboard made it hold together better, but as long as your learners have a target to aim at (i.e. the circle with the 5 in it), you're all good to go.

We scrunched up some pieces of packing paper to make the soft balls. We made about 15 of them, and put them in a box.

Before the experiment

Tell your learner that they will be dropping balls onto the piece of paper marked O. Have them take a few practice drops to understand what they will be doing.

If you think the task is either too easy or too hard for them, move the target. We started our practice drops at the top of the stairs (aiming at the floor below), and found that our kids (4 and 8) were not hitting any of the pages most of the time. No problem, we had them move down the stairs a bit, and dropped over the side of the stairs. For those of you that don't have stairs like this, you could always have them stand on a chair, and make the target smaller (for example, cut the pages in half before assembling the target) if they are easily hitting the middle rectangle each time. Basically, it should be a bit of a challenge for them to hit it, but you want them to be successful often enough that it is fun.

If they do this 50 times, how many times do they think that:

1. The ball lands on the paper marked with an O
2. The ball lands on one of the other pieces of paper
3. The ball does not land on any piece of paper

By "lands on the paper", we mean that we are interested in the first part of the target it hits. If it (say) hits one page, and then skids on to another, you want to count this as hitting the first page, and not the second. 

Note that these are reasonably subjective questions, but some things to think about are:

• The paper marked with O is the thing that they are aiming at
• There is only one page marked with an O, and eight without and O. If your learner's aim is spread out enough (i.e. on average your learner is a good dropper, but there is a lot of variation), then the ball should land on each of the nine pieces of paper about the same number of times. Therefore, the ball should land on the pages on the outside of the target eight times as often as it lands on the one marked with an O. That is, if the ball landed on the O page 5 times, then we would expect that the ball landed on other pages 5x8 = 40 times
• If their aim is better than this, then maybe the ball lands on the blank pages less than eight times as often than the O page. That is, if the ball landed on the O page 5 times, then we would expect that the ball landed on the blank pages less than 40 times

Running the experiment

Have your learner drop the balls many times (say, 50) from the same height (as economists, we are all in favor of providing some incentives for doing this, i.e. one m&m for every 5 successful drops). Have them record where the ball lands. Note that there are three possible outcomes:

1. Lands on the page marked with an O
2. Lands on one of the other pages
3. Misses all of the pages

Here is what our data look like:

That is, by the end of the experiment, our littles had hit the middle of the target 12 times, missed the middle but hit one of the other pages 26 times, and missed the target entirely 16 times.

Each row of this table before the total involved our kids throwing all of our balls off the stairs. When all balls were son the ground, they went down the stairs to pick them up, then started throwing again.

After the fourth iteration, the 4-year-old was trying to do trick shots off the wall (which I recorded as a miss), so it was a good time to stop.

Analyzing the data

Test your learner's predictions they made before the experiment against the results of the experiment. In particular, is the number of times the ball lands on one of the outside pages close to eight times the number of times it lands on the O page?

For our experiment, we had 12 balls land on the O page. If our learners were not particularly accurate, that means we would have expected 12x8 = 96 balls to land on the outside pages. In fact, only 26 did, and so from this we can conclude that our learners were more accurate than just spraying the target would predict.