Probability and Data

statisticalanalysis year11 advanced

Probability and data | NSW Curriculum Website

• MAV-11-09 solves problems involving probability in a variety of contexts
• MAV-11-10 displays and analyses datasets using summary statistics and graphical representations

📖 Prior Knowledge

Content	Prerequisite relationships
Data A	- Construct column graphs with a many-to-one scale → Represent data with a variety of graphs
Chance A	- List equally likely outcomes → List the sample space - Represent probabilities as fractions → Calculate theoretical probability
Fractions Decimals Percentages	- Find a percentage of a quantity → Represent data with a variety of graphs
Data Classification and Visualisation	- Represent data with a variety of graphs (revisited content)
Probability	- List the sample space (revisited content) - Calculate theoretical probability (revisited content)
Probability A	- Distinguish independent and dependent events (revisited content)
Probability B	- Represent data in Venn diagrams and 2-way tables (revisited content) - Use set notation for events (revisited content) - Calculate conditional probabilities (revisited content) - Describe mutually and non-mutually exclusive events → Calculate probabilities of compound events - Calculate probabilities of compound events (revisited content)
Data Analysis	- Group and represent data (revisited content)

Sets and set notation

• Define a set as a collection of objects, called the elements of the set, and represent a set using notation such as $\{2,3,5,7\}$ and $\{0,2,4,6,\dots \}$
• Use the notation $n(A)$ or $|A|$ to represent the number of elements in a finite set $A$
• Define the empty set as the set with no elements, denoted in set notation as $\varnothing$
• Use the notation $\overline{A}$, $A^{\prime }$ or $A^c$ to represent the complement of a set $A$ with respect to some universal set $U$
• Define A to be a subset of B if all the elements of $A$ are elements of $B$
• Define the intersection $A\cap B$ of sets $A$ and $B$ to be the set of elements that are in $A$ and in $B$
• Define the union $A\cup B$ of sets $A$ and $B$ to be the set of elements that are in $A$ or in $B$
• Define sets A and B to be disjoint if $A\cap B=\varnothing$, that is, they have no elements in common
• Use Venn diagrams in practical situations to represent and interpret sets that may intersect in various ways within a universal set
• Establish and use the rule $|A\cup B|=|A|+|B|-|A\cap B|$

Probability

• Define an experiment or a trial to be any procedure that can be infinitely repeated and has a well-defined set of possible outcomes known as the sample space, denoted $S$
• Identify an event $A$ as a subset $A$ of the sample space $S$
• Define the probability of each outcome to be $\frac{1}{|S|}$ when all the outcomes are equally likely and the probability of the event A to be $P(A)=\frac{|A|}{|S|}$
• Interpret the notation $\overline{A}$ to be the event ‘A does not occur’, interpret $A\cap B$ to be the event ‘A and B both occur’, and interpret $A\cup B$ to be the event ’$A$ or $B$ occurs’
• Use Venn diagrams to represent the relationship between events within the same sample space, including mutually exclusive events, that is, events that as subsets of the sample space are disjoint
• Establish and use the rules $P(\overline{A})=1-P(A)$ and $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
• Use arrays and tree diagrams to determine the outcomes and probabilities for multistage events

Conditional probability

• Define conditional probability as the probability that an event A occurs given that another event B has already occurred, and use the notation $P(A|B)$
• Examine conditional probability by restricting the sample space and event spaces in a Venn diagram, using a two-way table, a tree diagram and other arrays
• Establish that $P(A|B)=\frac{|A\cap B|}{|B|}$ when all outcomes are equally likely by restricting the sample space and event space, and hence $P(A|B)=\frac{P(A\cap B)}{P(B)}$, provided $|B|\ne 0$
• Use the formulas for $P(A|B)$ to solve practical problems involving conditional probability
• Define two events to be independent if the occurrence of one event does not affect the probability that the other event occurs
• Explain that two events A and B are independent means $P(A|B)=P(A)$ and $P(B|A)=P(B)$, and show algebraically that if one of these formulas is true, then the other is also true
• Use the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$, and the test $P(A|B)=P(A)$ for independence to prove that if two events are independent, then $P(A\cap B)=P(A)\times P(B)$, and to prove conversely that if $P(A\cap B)=P(A)\times P(B)$, then A and B are independent
• Solve practical problems involving independent events

Data

• Define a random variable as a variable whose value is the outcome of a random experiment
• Compare discrete random variables with continuous random variables, describe their differences, and give practical examples of each
• Organise finite datasets using a table or a spreadsheet, listing the values, frequency, relative frequency, cumulative frequency, and cumulative relative frequency
• Graph the frequency, relative frequency, and cumulative frequency histograms and polygons of datasets, using spreadsheets or graphing applications, and identify the mode and median from the graphs, and from tables
• Use the relative frequency to estimate the probability of results in experiments