Probability

Introduction to probability

Probability tells us how likely something is to happen. It's like a measure of chance.

The probability scale is a number line from 0 to 1 where we can place the probability of events occurring.

We use the letter $P$ to show probability.

• $P(A)$ means "the probability of event $A$ happening".
• $P(A')$ means "the probability of $A$ not happening".

For example, if $P(A) = 0.3$, then $P(A') = 0.7$.

Probability can be written in many ways (such as in the table).

How to calculate basic probability

Here is a general formula for calculating probability:

• Total outcomes: 6 (a dice has the numbers 1 to 6).
• Desired outcome: 1 (only one 4).

$P(4) = \frac{1}{6}$

• Total: $3 + 2 + 5 = 10$.
• Red $= 3$.

$P(\text{red}) = \frac{3}{10}$

$P(\text{blue}') = 1 - 0.2 = 0.8$

Relative and expected frequencies

Sometimes we can't calculate exact probabilities, so we do experiments instead and use the results to estimate. That's where relative frequency comes in.

$P(\text{red}) = \frac{25}{100} = 0.25 \qquad P(\text{blue}) = \frac{30}{100} = 0.30 \qquad P(\text{green}) = \frac{45}{100} = 0.45$

These are estimates, not exact probabilities. The more trials you do, the closer it gets to the real probability.

• This spinner is not fair.
• A fair spinner would be one that has equal probability for all outcomes.
• In this example, the spinner would be fair if the probability of each outcome was $\frac{1}{3}$ ($0.33$).
• Since this spinner is not fair, it is called a biased spinner.

Once you know the probability of something happening, you can use it to predict how often it should happen in future trials. This is called the expected frequency.

$\text{Expected frequency} = \frac{1}{6} \times 60 = 10$

Sometimes, outcomes can be random, meaning each outcome has a chance and there is no pattern. An expected frequency is therefore an average, not a promise: you may roll more or fewer than 10 sixes.

Probability of combined events

A combined event is when two or more things happen, like flipping two coins, or picking two balls one after another.

Sample space diagrams

For this question, we make a sample space diagram, listing every possible outcome. The shaded squares are the sums that equal 7.

$P(7) = \frac{\text{number of desired outcomes}}{\text{total}} = \frac{6}{36} = \frac{1}{6}$

Venn diagrams

Check the total first: $6 + 2 + 1 + 1 + 5 + 3 + 2 = 20$ people.

(a) Everyone inside the football circle plays football: $6 + 2 + 1 + 1 = 10$.

$P(\text{football}) = \frac{10}{20} = 0.5$

(b) Cricket and hockey is the overlap of those two circles, which includes the centre: $2 + 1 = 3$.

$P(\text{cricket and hockey}) = \frac{3}{20} = 0.15$

Tree diagrams

The probabilities on each pair of branches must add up to 1. The first day below 14°C has probability $1 - 0.35 = 0.65$. The temperature on the second day does not depend on the first day, so its branches have the same probabilities: $0.35$ and $0.65$.

To find the combined probabilities, we multiply the probabilities of the events occurring along the branches.

$P(\text{above 14°C on both days}) = 0.35 \times 0.35 = 0.1225$

Without replacement

Without replacement means that the ball that was selected will not be put back with the other balls for the next selection. So for the second ball:

• the denominator reduces by 1, because there are only 11 balls left;
• if a black ball was taken first, there are $x - 1$ black balls left and still $12 - x$ white balls;
• if a white ball was taken first, there are still $x$ black balls left and $12 - x - 1 = 11 - x$ white balls.

There are 2 possibilities here: we take the black ball and then the white, or the white and then the black. Hence, we add the two probabilities.

$\frac{x}{12} \times \frac{12 - x}{11} + \frac{12 - x}{12} \times \frac{x}{11} = \frac{x(12 - x)}{132} + \frac{x(12 - x)}{132} = \frac{2x(12 - x)}{132} = \frac{x(12 - x)}{66}$