Predict and test

Roll a dice 60 times. Exactly ten sixes? Probably not. But how far off is normal, and when should you suspect the dice?

So in 60 rolls of a fair dice you expect about

60 \times \tfrac{1}{6} = 10

sixes. Then you test it: run the experiment, record the counts in a table, and compare. The relative frequency (count ÷ trials) is the experiment's estimate of the probability.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

How many sixes?

What fraction of rolls should be a six?

You roll a fair dice 60 times. About how many sixes do you expect?About 30You can't say anythingAbout 10Exactly 6

TryTry it on the board and compare the result with your prediction.

Roll 60 times

Compare with others: did anyone get exactly 10? What was the furthest from 10?

Try it yourselfYour edits are saved in this browser.

Roll 60 times (press Run 10 six times). How many sixes did you get? Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Expected and observed

Here is how to predict and test.

Worked exampleBefore each step, say what you think comes next.

$P(6) = \tfrac{1}{6}$ The probability from the sample space.
$60 \times \tfrac{1}{6} = 10$ Expected count: probability × trials.
$\text{observed: } 7 \text{ sixes}$ Run the experiment and record the count.
$\tfrac{7}{60} \approx 0.12$ Relative frequency = count ÷ trials, the experiment's estimate.
$0.12 \text{ against } 0.17$ Close but not equal: a short experiment varies.

Now: how many heads in 50 flips?

Read Noah's reasoning. Is 7 instead of 10 surprising for a fair dice?

You flip a fair coin 50 times. How many heads do you expect?

Noah tests a dice. Click the line where his reasoning goes wrong.

In an experiment, a coin landed heads 13 times in 20 flips. What is the relative frequency of heads? (A decimal.)

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

Expected: 100 × 1/2 = 50 heads.

Need a hint?

Red is 3 of 8 equal parts. Run 80 spins and look at the expected count column.

Try it yourselfYour edits are saved in this browser.

Flip the coin 100 times. Compare the number of heads with the expected 50.

Use the board. This ticks when you get there.

A spinner is $\tfrac{3}{8}$ red. How many reds do you expect in 80 spins?

Spin this spinner (3 red, 5 blue sectors) at least 80 times. Compare the reds with the expected count.

Use the board. This ticks when you get there.

Two students flip a coin 10 times each. Ana gets 7 heads; Ben gets 5. Is Ana's coin unfair?Yes: 7 is not 5Not necessarily: 10 flips is too few to tellYes: Ben's is the fair one

Edge case: you roll a dice 50 times. What is the expected number of sixes?Exactly 8You can't expect anything because 50 is not a multiple of 6About 8.3 (50 × 1/6): you can expect a value you can never actually get

PractiseA few on your own, to check you have it.

Expected counts

A bag has 2 red and 8 blue counters. You take one and put it back, 40 times. How many reds do you expect?

A spinner landed on green 18 times in 60 spins. What is the relative frequency of green? (A decimal.)

Which result would make you most suspicious of a coin?52 heads in 100 flips6 heads in 10 flips900 heads in 1000 flips

Review: roll a fair dice. What is $P(\text{a number less than } 3)$ ? Simplest form.

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Plan an experiment

Then: how many spins of the 3-red, 5-blue spinner to expect 30 reds?

You want to expect 25 sixes from a fair dice. How many times should you roll it?

Predict and test

How many sixes?

Roll 60 times

Expected and observed

One change at a time

Expected counts

Plan an experiment

Students will be able to

Before class

Running the lesson

When students get stuck

Stretch and support

Where it leads

What each step is for