Relative frequency and probability

When an experiment is repeated many times, the share of the times an event happened comes near its probability.

• This share is called the relative frequency of the event.
• With it you can estimate a probability that cannot be calculated – for example, when you do not know what is in a bag.

• The frequency of an event is the number of times it happened.
• The relative frequency of an event is its frequency out of the number of times the experiment was done:

• $\displaystyle k$ – the number of times the event happened.
• $\displaystyle n$ – the number of times the experiment was done.
• The relative frequency is a number between 0 and 1, like a probability.
• It can be written as a fraction, a decimal or a percentage.

The example below calculates a relative frequency.
Noa tossed a coin 20 times.
It came up heads 12 times.

• The frequency of heads is 12.
• The relative frequency of heads: $\displaystyle {\frac{12}{20}}={\frac{3}{5}}={0.6}$.
• So 60% of the tosses were heads.

In the example below a die was thrown 1,000 times, and each throw was checked for a 6.

• The dashed line on the graph is the probability of a 6: $\displaystyle \frac{1}{6}\approx0.167$.
• The solid line is the relative frequency of 6 after each number of throws.

What the graph shows:

• After 10 throws, 4 of them were a 6: the relative frequency is $\displaystyle {\frac{4}{10}}={0.4}$ – far from the probability.
• After 100 throws, 22 of them were a 6: $\displaystyle {\frac{22}{100}}={0.22}$ – already nearer.
• After 1,000 throws, 160 of them were a 6: $\displaystyle {\frac{160}{1000}}={0.16}$ – very near $\displaystyle \frac{1}{6}$.

The more experiments are done, the less a single one changes the relative frequency:

• After 10 throws, one more 6 changes the relative frequency by a tenth, $\displaystyle \frac{1}{10}$.
• After 1,000 throws, one more 6 changes it by only $\displaystyle \frac{1}{1000}$.

So:

• In a few experiments, a few “lucky” outcomes in a row take the relative frequency far from the probability. A coin tossed 4 times can come up heads all 4 times – a relative frequency of 1.
• In many experiments, such runs hardly change anything, and the relative frequency stays near the probability.
• The relative frequency is not exactly the probability, even after many experiments – only near it.

When the outcomes are not equally likely and there is nothing to count, the probability cannot be calculated. Then the experiment is done many times, and the relative frequency is an estimate of the probability.

The example below estimates the probability of an event when a drawing pin is thrown.
A drawing pin thrown on a table lands either point up or on its side.
Yoav threw a drawing pin 500 times.
It landed point up 310 times.

• The relative frequency: $\displaystyle {\frac{310}{500}}={0.62}$.
• So the probability that the pin lands point up is about 0.62.
• If Yoav throws the pin another 1,000 times, he can expect it to land point up about $\displaystyle {1000\cdot0.62}={620}$ times.

The example below estimates how many red marbles are in a bag.
A bag holds 20 marbles, red and blue, and you cannot see inside it.
A marble is drawn, its colour written down, and it is put back into the bag – 200 times.
A red marble came out 152 times.

• The relative frequency of red: $\displaystyle {\frac{152}{200}}={0.76}$.
• So the probability of drawing a red marble is about 0.76.
• The probability of red is the number of red marbles out of 20, so about 0.76 of the marbles are red: $\displaystyle {0.76\cdot20}={15.2}$.
• The estimate: about 15 red marbles and 5 blue ones in the bag.

Find the relative frequency.
Write it as a fraction in its lowest terms and as a decimal.

A basketball player took 40 free throws and scored with 28 of them.
What is the relative frequency of scoring?

Shira threw a die 50 times, and 3 came up nine times.
What is the relative frequency of 3?

A factory checked 250 light bulbs, and 10 of them were faulty.
What is the relative frequency of a faulty bulb?
Write it as a percentage too.

A coin is tossed.
The event: heads.
The probability of the event is $\displaystyle \frac{1}{2}$, and it is marked on the graph.
Toss the coin 10 times, then 100 times more, then 1,000 times more.
After each stage, write how many tosses there have been, the relative frequency of heads, and how far it is from $\displaystyle \frac{1}{2}$.
What happens to that distance as the number of tosses grows?

A die is thrown.
The event: an even number.

Find the probability of the event.

Throw the die 1,000 times.
Is the relative frequency of the event near the probability you found in part a?

A bag holds 10 marbles, red and blue, and you cannot see inside it.
In each experiment one marble is drawn, its colour written down, and it is put back into the bag.

Draw a marble 10 times.
How many red marbles do you think are in the bag?

Draw a marble 1,000 times more.
What is your estimate now?

Which of the two estimates is better? Explain why.

A spinner has three parts: red, blue and green. Whether the parts are the same size is not known.
The spinner was spun 200 times, with these results:

Find the relative frequency of each colour.

Which part of the spinner is probably the largest? Explain.

About how many times will the arrow stop on green, if the spinner is spun 1,000 times?

What is wrong with each claim? Explain.

Dana threw a die 12 times, and no 6 came up.
She says: “On my die, 6 comes up less often than any other number.”

Yossi tossed a coin 4 times, and it came up heads 3 times.
He says: “The probability of heads is $\displaystyle \frac{3}{4}$.”