The complement and relative frequency

The complement of $\displaystyle A$ is the event “$\displaystyle A$ does not happen”: all the outcomes that are not in $\displaystyle A$. It is written $\displaystyle \overline{A}$ – the letter $\displaystyle A$ with a bar over it.

• When a die is thrown, $\displaystyle A$ – “a 6 comes up”: the outcome 6.
• $\displaystyle \overline{A}$ – “a 6 does not come up”: the outcomes 1, 2, 3, 4, 5.
• Every outcome is in exactly one of them: in $\displaystyle A$ or in $\displaystyle \overline{A}$.

The probability of an event and the probability of its complement add up to 1:

• Why: if $\displaystyle A$ holds $\displaystyle k$ outcomes out of $\displaystyle n$, $\displaystyle \overline{A}$ holds all the rest – $\displaystyle n-k$ outcomes. And $\displaystyle {\frac{n-k}{n}}={1-\frac{k}{n}}$.
• For a die: $\displaystyle {P(A)}={\frac{1}{6}}$, so $\displaystyle {P(\overline{A})}={1-\frac{1}{6}}={\frac{5}{6}}$.

The example below finds the probability of the complement when only the probability of the event is given, with no outcomes to count.

The weather forecast said that the probability of rain tomorrow is 0.3.
The event $\displaystyle A$: it rains tomorrow.
The complement, $\displaystyle \overline{A}$: it does not rain tomorrow.

A probability can be written as a fraction, a decimal or a percentage – it is the same number:

• $\displaystyle {\frac{1}{4}}={0.25}={25\%}$;
• $\displaystyle {\frac{3}{10}}={0.3}={30\%}$;
• a certain event: $\displaystyle {1}={100\%}$; an impossible event: $\displaystyle {0}={0\%}$.
• A forecast of “a 30% chance of rain” is a probability of 0.3, and the probability that it does not rain is 70%.

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

• The relative frequency: the number of times the event happened, divided by the number of times the experiment was done.

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.

• 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 reason: the more throws there are, the less a single throw changes the relative frequency.

• After 10 throws, one more 6 changes the relative frequency by a tenth.
• After 1,000 throws, one more 6 changes it by only a thousandth.
• The relative frequency is not exactly the probability – only near it.

When there are no equally likely outcomes to count, the relative frequency in data that were collected is an estimate of the probability.

The example below estimates the probability that a pupil chosen at random walks to school.
At random – by a draw, so that every pupil has the same chance of being chosen.
At a big school, 200 pupils were asked how they get to school.
The answers, in a frequency table:

• The relative frequency of “on foot”: $\displaystyle {\frac{84}{200}}={0.42}={42\%}$.
• So the probability that a pupil of the school, chosen at random, walks to school is about 0.42.
• If the pupil is chosen at random from the 200 who answered, the probability is exactly 0.42: each pupil is an outcome, and 84 of them walk.

When an experiment is repeated $\displaystyle n$ times, an event $\displaystyle A$ is expected to happen about $\displaystyle n\cdot P(A)$ times:

• The reason: the relative frequency is near $\displaystyle P(A)$, that is, the event happens in about a $\displaystyle P(A)$ share of the times.
• A die is thrown 600 times: a 6 is expected about $\displaystyle {600\cdot\frac{1}{6}}={100}$ times – maybe 93, maybe 108.
• At another, similar school with 500 pupils: about $\displaystyle {500\cdot0.42}={210}$ of them are expected to walk.

Find the probability of the complement.

The probability that Alon scores with a free throw is 0.7.
What is the probability that Alon misses?

A bag holds 8 marbles, and 3 of them are green.
One marble is drawn without looking.
What is the probability that the marble is not green?

The probability of winning a prize in the sports club’s raffle is 2%.
What is the probability of not winning? Write the answer as a percentage.

Write the first two probabilities as percentages.
Write the last probability as a fraction in its lowest terms.

A die is thrown.
The event: a number greater than 4 comes up.

Find the probability of the event.

Throw the die 10 times, then 1,000 times more.
After each stage, write the relative frequency of the event.
When is it nearer the probability you found in part a? Explain why.

In a survey, the 160 eighth-graders of a school were asked how many brothers and sisters they have.
The answers, in a frequency table:

One of the 160 pupils is chosen at random.

What is the probability that the pupil has no brothers or sisters?

What is the probability that the pupil has 2 or more brothers and sisters?

Write the probability from part b as a percentage.

Find about how many times the event is expected to happen.

A die is thrown 300 times.
How many times is an even number expected to come up?

A drawing pin thrown on a table lands either point up or on its side.
Many experiments found that the probability of landing point up is about 0.6.
How many times is the pin expected to land point up in 450 throws?

Another, similar school has 400 pupils.
By the survey in Exercise 4, how many of them are expected to have no brothers or sisters?

What is wrong with each claim? Explain.

The probability of a 6 when a die is thrown is $\displaystyle \frac{1}{6}$.
Amit says: “If I throw a die 6 times, a 6 will come up exactly once.”

Tamar tossed a coin 5 times, and every toss came up tails.
Tamar says: “The next toss will almost surely be heads.”