Level 3MathsAQAMathematical Studies (Core Maths)Paper 2B: Critical Path & Risk AnalysisRevision NotesProbability & ExpectationProbability & ExpectationProbabilities & Expected Outcomes

Probabilities & Expected Outcomes (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Exam code: 1350

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Basic Probability

What do I need to know about probability?

  • The language used in probability can be confusing so here are some definitions of commonly misunderstood terms

    • An experiment is a repeatable activity that has a result that can be observed or recorded; it is what is happening in a question

    • An outcome is the result of an experiment

  • All possible outcomes can be shown in a sample space

    • A sample space is a list or a table of all possible outcomes of an experiment

    • For example, the sample space diagram below shows two fair four-sided spinners being spun

      • The outcome is the product of the two sides the spinners land on

Sample space for two spinners being spun, and their scores being multiplied

  • An event is an outcome or a collection of outcomes; it is what we are interested in happening

    • An event could be more than one outcome

    • E.g. For the spinners above

      • The event “the product is -2” has one outcome 

      • The event “the product is negative” has 6 outcomes

What probability terminology and notation do I need to know?

  • Not A means the event A does not happen

    • You may see this written as straight A apostrophe

  • Probability notation that you should know includes:

    • straight P open parentheses straight A close parentheses equals 0.6 “The probability of event straight A happening is 0.6”

    • straight P open parentheses straight A apostrophe close parentheses equals 0.4 “The probability of event straight A not happening equals 0.4”

    • straight P open parentheses straight X less or equal than 4 close parentheses equals 0.4 “The probability of straight X being less than four is 0.4”

    • straight n open parentheses straight A close parentheses is the number of outcomes that are included in event A

      • An event can have one or more than one outcome

How do I solve probability questions?

  • To find the probability of a particular event occurring

    • straight P open parentheses " success " close parentheses equals fraction numerator number space of space ways space to space get space " success " over denominator total space number space of space outcomes end fraction

      • It is important to understand that the above only applies if all outcomes are equally likely

    • straight P open parentheses not space straight A close parentheses equals straight P open parentheses straight A apostrophe close parentheses equals 1 minus straight P open parentheses straight A close parentheses

      • The probability of "not A" is the complement of the probability of “straight A”

Worked Example

100 skydivers took part in an all-day charity event, with the altitude of the aeroplane at which they jumped from summarised in the histogram below.

Histogram showing frequency density, and altitude

(a) Use the histogram to find the probability that a randomly chosen skydiver jumped from the aeroplane at an altitude

(i) between 14 000 and 16 000 feet.

Answer:

Notice that:
The altitude scale is given in thousands of feet
The histogram uses frequency density

For the bar between 14 and 16

frequency density = 7, class width = 16 - 14 = 2
frequency = 7 x 2 = 14

Divide the number of skydivers who jump between 14 000 and 16 000 feet by the total number of skydivers

14 ÷ 100

0.14

(ii) between 16 000 and 20 000 feet.

Answer:

Between 16 000 and 20 000 feet comprises two groups
Work out the frequencies separately

16 000 to 18 000 feet

  frequency density = 12, class width = 18 - 16 = 2
frequency = 12 x 2 = 24

18 000 to 20 000 feet

frequency density = 9, class width = 20 - 18 = 2
frequency = 9 x 2 = 18

Find the total frequency of the two groups and divide by the total number of skydivers

(24 + 18) ÷ 100 = 42 ÷ 100

0.42

(b) Estimate the probability that a randomly chosen skydiver jumped from the aeroplane at an altitude between 13 000 and 15 000 feet.

Answer:

Parts of two groups are required, so use interpolation on both

15 000 feet is the midway point of the 14 000 to 16 000 feet group, so divide the frequency from (a) (i) by 2

14 ÷ 2 = 7

Find the total frequency of the 10 000 to 14 000 feet group

frequency density = 6, class width = 14 - 10 = 4
frequency = 6 x 4 = 24

13 000 to 14 000 is the last quarter of that group, so a quarter of the frequency would be expected to be in it
Divide the frequency of that group by 4

24 ÷ 4 = 6

Find the total frequency from 13 000 to 15 000 feet and divide by the total number of skydivers

(7 + 6) ÷ 100 = 13 ÷ 100

0.13

Remember that this value is an estimate because you have interpolated

Examiner Tips and Tricks

Most probability questions are in context so can be long and wordy; go back and re-read the question whenever you need to.

Try to get immersed in the context of the question to help understand a problem.

Expected Outcomes

What is expected frequency?

  • Expected frequency refers to the number of times that you would expect a particular outcome to occur when repeating a trial numerous times

  • The theoretical probability of that outcome will need to be known

    • Or an estimate of it, from relative frequency

How do I find expected frequency?

  • If the probability of a particular outcome is p and there are n trials then:

    • The expected number of occurrences of that outcome from the n trials is np

    • You simply multiply the number of trials by the probability of the particular outcome

  • Note that this does not mean that there will exactly np occurrences

    • But if the experiment (of n trials) was repeated over and over again we would expect the number of occurrences to average out to be np

Examiner Tips and Tricks

Exam questions will not necessarily use the phrase expected frequency or expected outcomes so think about the information given carefully.

If a question mentions repeatedly carrying out a trial, and a number of occurrences is requested (rather than a probability), expected frequency is involved.

Worked Example

There are 6 blue, 4 red and 5 yellow counters in a bag.  One counter is drawn at random and its colour noted.  The counter is then returned to the bag.

(a) Find the probability that a counter drawn from the bag is yellow.

Answer:

Identify how many yellow counters there are in the bag

5 yellow counters

Determine the total number of counters in the bag

6 + 4 + 5 = 15

P(Yellow)bold equals bold 5 over bold 15 bold equals bold 1 over bold 3

(b) How many times would you expect a yellow counter to be drawn if the experiment is repeated 300 times?

Answer:

This is expected frequency so multiply the number of trials (n) by the probability (p)

300 cross times 1 third equals 100

We would expect 100 yellow counters

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Author
Reviewer
Naomi C

Author: Naomi C

Expertise: Maths Content Creator

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.