The rules of probability

The probability of an event is the sum of the probabilities of its outcomes. Rules follow from this that hold for every event, even when the outcomes are not equally likely:

• $\displaystyle {P(U)}={1}$ – the certain event; $\displaystyle {P(\emptyset)}={0}$ – the impossible event.
• If all the outcomes of $\displaystyle A$ are also in $\displaystyle B$ – written $\displaystyle A\subseteq B$ – then $\displaystyle P(A)\le P(B)$: $\displaystyle B$ has all the outcomes of $\displaystyle A$, and perhaps more.
• The complement: $\displaystyle A$ and $\displaystyle \overline{A}$ together hold all the outcomes, so

• Disjoint events $\displaystyle A_1, A_2, \dots, A_n$ – no two of them have an outcome in common: the probability of the union is the sum of the probabilities.

In the sum $\displaystyle P(A)+P(B)$ the outcomes of the intersection are counted twice – once in $\displaystyle A$ and once in $\displaystyle B$. So they are taken away once:

For disjoint events $\displaystyle {P(A\cap B)}={0}$, and the sum is back.

In a certain town 60% of the families subscribe to a newspaper, 45% subscribe to a streaming service, and 25% subscribe to both.
A family is chosen at random.
$\displaystyle N$ – subscribes to a newspaper; $\displaystyle S$ – subscribes to a streaming service.

• To at least one of them: $\displaystyle {P(N\cup S)}={0.6+0.45-0.25}={0.8}$.
• To neither: $\displaystyle {P(\overline{N}\cap\overline{S})}={1-P(N\cup S)}={0.2}$.
• Only to a newspaper: $\displaystyle {P(N\cap\overline{S})}={P(N)-P(N\cap S)}={0.6-0.25}={0.35}$.

When a point is chosen at random inside a shape, and each part of the shape has a chance in keeping with its area, the probability of hitting a part is the part’s area divided by the area of the whole shape.

A dart hits a round target of radius 30 cm at random.
At the centre of the target is a coloured disc of radius 10 cm.

The probability of hitting the coloured disc:

The coloured disc’s radius is a third of the target’s radius, but the probability is only a ninth.

The relative frequency of an event is the share of the times it happened out of all the times the experiment was repeated. When an experiment is repeated many times, the relative frequency comes close to the probability – this is the law of large numbers. So when there is no way to work out a probability, it is estimated by the relative frequency of many repeats.

A fair die is thrown. The probability of an even number greater than 3 is $\displaystyle \frac{1}{3}$. Throw the die 10 times, 10 times more, and then 100 and 1000 times:

• After 10 throws the relative frequency can be very far from $\displaystyle \frac{1}{3}$, and every run of 10 throws gives another number.
• After thousands of throws the line stays close to the dashed line, $\displaystyle \frac{1}{3}$.

A single experiment or a few say little about the probability; a large number of experiments does.

$\displaystyle {P(A)}={0.5}$, $\displaystyle {P(B)}={0.4}$, $\displaystyle {P(A\cup B)}={0.75}$.

Work out $\displaystyle P(A\cap B)$.

Work out $\displaystyle P(A\cap\overline{B})$ – the probability that $\displaystyle A$ happens and $\displaystyle B$ does not.

Work out the probability that neither $\displaystyle A$ nor $\displaystyle B$ happens.

The probability of $\displaystyle B$ is twice the probability of $\displaystyle A$.
$\displaystyle {P(A\cap B)}={0.1}$, $\displaystyle {P(A\cup B)}={0.8}$.

Write $\displaystyle {P(A)}={x}$, and find $\displaystyle x$ with the probability of the union.

What is the probability that only $\displaystyle B$ happens?

A square dartboard has sides of 20 cm.
A circle of radius 10 cm is drawn on the board, touching its four sides.
A dart hits the board at random.

What is the probability that the dart lands outside the circle? Round to three decimal places.

$\displaystyle {P(A)}={0.7}$, $\displaystyle {P(B)}={0.6}$.

Can $\displaystyle A$ and $\displaystyle B$ be disjoint events? Explain with the rules.

Explain why $\displaystyle P(A\cap B)\ge0.3$.