Sixth Sense
Making Sense of Probability
The mathematics of probability at Level 3 is often technically quite simple – the heavy artillery of integration is not needed until the later A2 modules – but the language and approaches used often cause problems. Many students bring significant misconceptions from their GCSE, and earlier, study of probability, and so at the start of any work at Level 3 in probability, it is worth asking a range of questions to assess the security of students’ fundamental model of probability, before introducing the new ideas such as probability distributions and hypothesis testing. Good questions will also ensure students’ familiarity with some key tools that help to represent probability problems, and thereby support their understanding of concepts in probability.
Questions about familiar contexts can help explore students’ probabilistic intuition. For example, a question such as
The probability that I roll a multiple of 3 on a fair die is ^{1}⁄_{3}. I have rolled the fair die twenty times without getting a multiple of 3. Is the probability that I will roll a multiple of 3 on the twentyfirst roll
a. considerably greater than ^{1}⁄_{3}?
b. approximately equal to ^{1}⁄_{3}?
c. considerably less than ^{1}⁄_{3}?
emphasises to students that the probability THEY assign to a situation is the number that THEY choose to represent THEIR degree of belief that something will (or will not) happen: the probability is not “in the die”, it’s “in the observer’s head”.
Contrasting this scenario with
I have a list of hard mental arithmetic questions. When I ask my pupils these questions, about a third of the questions are answered correctly. I have asked James twenty of these questions. He has answered none of them correctly. Is the probability that James answers the next question I ask him correctly
a. considerably greater than ^{1}⁄_{3}?
b. approximately equal to ^{1}⁄_{3}?
c. considerably less than ^{1}⁄_{3}?
will ensure that students think about how additional information (either implicit or explicit) alters their degree of belief about a scenario, and hence the probability that they choose to assign to the possible outcomes. These scenarios also help students to refine their understanding of the independence of events.
Similarly, useful and important discussions will be prompted by comparing two statements such as
a. “Robin is in my year 9 class. Half the boys in my year 9 class have ginger hair. Therefore the probability that Robin has ginger hair is 0.5“
and
b. “My birthday is in November. Half the days in November last year were wet. Therefore the probability that my birthday will be wet next year is 0.5“.
Discussion of the scenario
Blaise thinks that the probability that a drawing pin will land headsidedown when it is flicked in the air is ^{1}⁄_{3}. He flicks ten drawing pins in the air and six of them land headsidedown.
a. Is Blaise wrong about his assessment of the probability?
Blaise then goes on to flick one thousand drawing pins in the air and five hundred and eighty seven of them land headsidedown.
b. Does this additional evidence change your opinion in a?
René says “The probability of something is a fraction of times it has happened. Therefore the probability of a drawing pin landing headsidedown was ^{6}⁄_{10}; then it changed to ^{587}⁄_{1000} in the second experiment. It does not make sense to talk about the probability that something will happen because you don’t know about the future
will lead to a valuable debate around experimental probability and the use of probability to quantify uncertainty.
Further exploration of probability through traditional questions such as
You are wandering down the street in your local town or city. Place the following events in the order of their probability, from lowest to highest:
a. The next person I see is blonde
b. The next person I see is female
c. The next person I see is both blonde and female
d. The next person I see is either blonde or female (or both)
e. The next person I see is either a blond male or is female
will prompt discussion of mutually exclusive and independent events, and also the probabilities of combined events. This discussion could be facilitated by using a card sequencing activity for the probabilities of the five chosen events. It would also lead naturally to the use of Venn diagrams as a representation tool of probabilities: surely we don’t have to wait until September 2017 and the revised AS/A Level Mathematics to revitalise the use of Venn diagrams? It is interesting to note that, according to the Adventures of the Black Square exhibition currently at the Whitechapel Gallery in London, Venn diagrams were banned by the military junta in Argentina in the 1970s because they promoted (it was argued) overlapping ideologies and collaborative working!
Thinking “Venn diagrammatically” might also help students as they consider statements such as
a. The probability it will rain at some time tomorrow is ^{2}⁄_{5}; the probability that it will be dry all day tomorrow is ^{5}⁄_{8}.
b. The probability it will rain at some time tomorrow is ^{2}⁄_{5}; the probability that I will forget to do my maths homework tomorrow is ^{5}⁄_{8}.
c. The probability that it will rain tomorrow is ^{2}⁄_{5}; the probability that there will be torrential rain tomorrow is ^{5}⁄_{8}.
As students become more familiar with the conceptual model that probabilities measure THEIR degree of belief in, or knowledge about, a situation, so it will be easier for them to understand, discuss and resolve some of the classic “pseudoparadoxes” of probability, and to explain with confidence correct interpretations of risk and likelihood in circumstances such as medical testing and trial by jury. We shall look at these in the next issue.
Image credit
Page header by Richard Gray (adapted), some rights reserved
