Probability and Data

Probability and data is examined every year, usually as one or two multiple-choice questions plus a 2–4 mark Section II question on conditional probability or a Venn diagram. It also sets up Year 12 Random variables, which is one of the most heavily weighted topics in the course.

• Sets: set notation, $n(A)$, the empty set, complement, subset, union, intersection, disjoint sets, Venn diagrams, and $n(A\cup B)=n(A)+n(B)-n(A\cap B)$.
• Probability: sample spaces and events, equally likely outcomes, the complement and addition rules.
• Conditional probability: $P(A\mid B)$ from Venn diagrams, two-way tables, tree diagrams and arrays, and the formula.
• Independence: what it means and the test $P(A\cap B)=P(A)P(B)$.
• Data: random variables (discrete and continuous), frequency tables with relative and cumulative relative frequency, graphs, and relative frequency as an estimate of probability.

Topics from the old syllabus such as box plots, quartiles and bivariate data (correlation and regression) are not part of this focus area, so don't over-study them.

A set is a collection of elements, written in braces: $\{2,3,5,7\}$. Key notation:

$A$ and $B$ are disjoint if $A\cap B=\varnothing$. The universal set $U$ contains everything under consideration; draw it as the rectangle around a Venn diagram.

The intersection is subtracted because it was counted twice. When filling in a Venn diagram, always start with the intersection and work outwards.

Worked example: a Venn diagram and a conditional probability

In a group of 50 people, 28 drink tea, 23 drink coffee, and 9 drink neither. One person is chosen at random. Find the probability that they drink tea, given that they drink coffee.

The Venn diagram has 18 tea only, 10 both, 13 coffee only and 9 neither (total 50). Given coffee, the sample space shrinks to the 23 coffee drinkers:

An experiment has a sample space $S$ of possible outcomes; an event $A$ is a subset of $S$. When all outcomes are equally likely,

$A^{\prime }$ means "$A$ does not occur", $A\cap B$ means "$A$ and $B$ both occur", and $A\cup B$ means "$A$ or $B$ (or both) occurs".

If $A$ and $B$ are mutually exclusive (disjoint), $P(A\cap B)=0$ and the addition rule simplifies to $P(A)+P(B)$.

The complement rule is the fastest way to handle "at least one": $P(\text{at least one})=1-P(\text{none})$.

$P(A\mid B)$ is the probability of $A$ given that $B$ has occurred. Knowing $B$ happened restricts the sample space to $B$:

Rearranged, $P(A\cap B)=P(B)\times P(A\mid B)$. This is exactly what you do when you multiply along the branches of a tree diagram: the second-stage branches are conditional probabilities.

Use whichever representation fits the information:

• Venn diagram for counts or probabilities of overlapping events.
• Two-way table when two categories are cross-classified.
• Tree diagram for multi-stage experiments, especially without replacement.
• Array (grid) for two dice or two spinners.

Worked example: a tree diagram in reverse

A factory has two machines. Machine A makes 60% of the parts, and 3% of its parts are faulty. Machine B makes the rest, and 5% of its parts are faulty. A part is chosen at random.

(a) Find the probability that it is faulty.

(b) Given that the part is faulty, find the probability it came from machine A.

Machine A makes 60% of the parts but only about 47% of the faulty ones, because its fault rate is lower. In part (b), the numerator is one branch and the denominator is the total from part (a).

$A$ and $B$ are independent if one happening does not change the probability of the other: $P(A\mid B)=P(A)$ (and then also $P(B\mid A)=P(B)$). Substituting into the conditional probability formula gives the test:

Example. Of 160 students, 64 play a sport, 40 are in the band, and 16 do both. Then $P(S)=0.4$, $P(B)=0.25$ and $P(S\cap B)=0.1$. Since $0.4\times 0.25=0.1$, the events are independent: being in the band makes no difference to the chance of playing a sport.

Independent is not the same as mutually exclusive. If two events with non-zero probabilities are mutually exclusive, then $P(A\cap B)=0\ne P(A)P(B)$, so they cannot be independent. Knowing one happened tells you the other did not.

A random variable is a variable whose value is the outcome of a random experiment.

• Discrete: takes separate, countable values. Examples: the number of heads in 5 tosses, the number of customers in an hour, goals in a match.
• Continuous: can take any value in an interval, so it is measured, not counted. Examples: height, time, mass, temperature.

Frequency tables

Organise a data set in a table with these columns:

• Frequency $f$: how many times each value occurs.
• Relative frequency: $\frac{f}{\text{total}}$, the proportion of the data.
• Cumulative frequency: the running total of $f$.
• Cumulative relative frequency: the running total of relative frequency, ending at 1.

Example. The goals scored in 25 matches:

• The mode is 1 goal (highest frequency).
• The median is the 13th score. The cumulative frequency reaches 12 at 1 goal and 18 at 2 goals, so the median is 2. On a cumulative relative frequency table, it is the first value where the total passes 0.5.
• The estimated probability that a match has at least 3 goals is $\frac{5+2}{25}=0.28$.

Display discrete data with a frequency or relative frequency histogram (columns centred on each value), and cumulative frequency with an ogive or a cumulative frequency step graph.

Relative frequency as probability

Relative frequency from an experiment estimates the probability of an outcome. The estimate gets more reliable as the number of trials increases. The expected number of times an outcome occurs in $n$ trials is $n\times P$.

• Double counting the intersection when filling in a Venn diagram. Fill it in first.
• Dividing by the whole sample instead of $n(B)$ or $P(B)$ in a conditional probability.
• Confusing $P(A\mid B)$ with $P(B\mid A)$. Read "given" carefully: what comes after "given" goes in the denominator.
• Adding probabilities for events that overlap without subtracting the intersection.
• Treating draws without replacement as independent.
• Saying mutually exclusive events are independent.
• Calling a measured quantity such as time "discrete" because the data were rounded.

• Define your events with letters at the start ("Let $F$ = the part is faulty") and write the probability statement you are finding before you calculate.
• A labelled tree diagram or Venn diagram often earns the first mark on its own.
• Determine whether or justify independence: calculate $P(A)P(B)$ and $P(A\cap B)$, compare them, and write a conclusion sentence.
• Leave probabilities as exact fractions unless a decimal is asked for. Simplify fractions.
• For "at least one" questions, use the complement.

• Random variables (Year 12): discrete probability distributions with expected value and variance, then continuous random variables and the normal distribution.
• Working with functions (Year 11): set notation and interval notation are used for domains and ranges.
• Integral calculus (Year 12): probabilities for continuous random variables are areas under a curve.