Relative frequency in the long run

Data and chance, grade 6 Lesson 8 of 9Learning

Relative frequency in the long run

The dreidel

A dreidel – the Hanukkah spinning top – has four faces, with a Hebrew letter on each: Nun, Gimel, Hei, Pei. When the dreidel falls, the letter on top is the outcome.

A straight, symmetric dreidel has no face it tends to fall on.
So each of the four letters has the same chance: $\displaystyle \frac{1}{4}$ , that is 25%.

Ten spins

Noa spun the dreidel 10 times. This is how the letters came up:

Nun came up 5 times, and Gimel did not come up at all.
The relative frequency of Nun: $\displaystyle {\frac{5}{10}}={0.5}$ – twice the chance, $\displaystyle {\frac{1}{4}}={0.25}$ .
In only ten spins luck has a large effect: one letter can come up a lot, and another not at all.

A pupil, a group and the class

In Noa’s class every pupil spun a dreidel 10 times. Then each group of four pupils put its results together, and in the end the whole class put them together:

	Spins	Nun	Gimel	Hei	Pei	Relative frequency of Nun
Noa	10	5	0	3	2	$\displaystyle {\frac{5}{10}}={0.5}$
Noa’s group	40	13	7	11	9	$\displaystyle {\frac{13}{40}}={0.325}$
The whole class	300	79	71	77	73	$\displaystyle \frac{79}{300}\approx0.26$

The more spins, the closer the relative frequency of Nun to the chance, 0.25.
The other letters too: in the whole class each of them came up in about a quarter of the 300 spins, that is about 75 times.

1,000 spins

On a computer it is easy to spin a dreidel many times. These are the results of 1,000 spins:

Each letter came up about 250 times – a quarter of 1,000.
The relative frequency of each letter is very close to 0.25, but not exactly equal to it.
The reason: when there are many spins, one spin changes the relative frequency only a little, and luck hardly shows.

Estimating a chance by an experiment

The chance of something that is not symmetric is estimated by its relative frequency in many experiments.
The example below estimates the chance that a drawing pin lands on its back, point up.
A pin was dropped 500 times, and it landed on its back 310 times.

The relative frequency: $\displaystyle {\frac{310}{500}}={\frac{62}{100}}={0.62}$ .
So the chance that the pin lands on its back is about 0.62, that is about 62% – more than a half.
The more times it is dropped, the better the estimate.

Exercise 1

Spin the dreidel 10 times, then 100 more times, and then 1,000 more times.
After each stage write down the relative frequency of Gimel.
When is it closest to $\displaystyle \frac{1}{4}$ ?

Exercise 2

The pupils of class 6B tossed coins and wrote down how many times heads came up:

	Tosses	Heads
Yossi	10	7
Yossi’s group	50	29
The whole class	400	206

What is the relative frequency of heads in the whole class? Write it as a decimal.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Whose relative frequency of heads is closest to $\displaystyle \frac{1}{2}$ – Yossi’s, the group’s or the class’s? Why?

Exercise 3

A bottle cap was dropped 200 times, and it landed on its open side 70 times.

Estimate the chance that the cap lands on its open side. Write it as a decimal.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

About how many times will the cap land on its open side if it is dropped 1,000 times?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A symmetric dreidel is spun 200 times.
About how many times will Gimel come up?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Dan spun his dreidel 400 times.
Nun came up 160 times, and each of the other three letters 80 times.
Is Dan’s dreidel symmetric? Explain with the relative frequencies.