Roll a prize

This task is about how likely things are to happen.

Question

People roll a fair dice to see if they win a prize at a school gala.

Roll a 1, 2 or 3	Roll a 4	Roll a 5	Roll a 6
No prize	Win a biscuit	Win a biscuit	Win an ice cream

James rolls the dice. Which of these is true? (select one)

- James is more likely to win a biscuit than an ice cream.
- James is more likely to win an ice cream than a biscuit.
- James is just as likely to win a biscuit as an ice cream.

Explain your answer

Question

Roll a 1, 2 or 3	Roll a 4	Roll a 5	Roll a 6
No prize	Win a biscuit	Win a biscuit	Win an ice cream

Elle has a turn rolling the dice. Which of these is most likely? (select one)

- Elle wins a prize.
- Elle does not win a prize.
- Elle is just as likely to win a prize as she is to not win one.

Explain your answer

Teaching and learning:

This resource looks at uncovering students' conceptions and misconceptions in probability reasoning. The explanation is more revealing of students' thinking than the multiple choice response.

Diagnostic and formative information:

Common incorrect answers
There are many well known misconceptions about probability. Examples of these follow.

For more information click on the link Probability concept map: Common misconceptions. The letters [A], [B] or [C] in each example below is the response the student chose in the multiple choice part of that question.

	Likely misconception
a) b)	Equiprobability The student sees events as equally likely even when they have different probabilities. Because there is a biscuit and an ice cream in the competition [C]. Because there is 3 chances of getting a prize and there is only one chance of getting no prize [B]. I looked at the chart. There are more prizes than no prize [B]. Because they would all get one [prize] [A].
a) b)	Outcome approach Some students may respond to situations involving chance by stating that you just can't tell anything when it comes to probability. [It] depends on what he rolls. He might not always win [C]. Because you can't make the dice land on a 5 or a 4, it just lands on what it lands on. Because you could get any number [C]. Elle may win a prize, she may not win a prize. The best thing is she had a go. She may have better luck next time [C].
a) b)	Regency (a form of representative bias) Students often think that if a particular event has happened more often than they expect, then it will be less probable to occur (negative regency) or more likely to occur (positive regency). He already has 2 biscuits so it is more likely that he will get another one [A - positive regency]. Coz she might not have won one yet [C – negative regency].
a) b)	Lack of independence - Influence Students believe that previous events can be influenced by a range of factors. For example, the outcome of a coin toss may depend on what the previous toss was. This assumes that the starting position of the arrow positively influences where it finishes up; James will always try and get the most valuable thing [B - influenced by value of item]. The ice cream is [worth] more than a biscuit and if it was a hot day an ice cream is better than a biscuit [B – influenced by the value or the weather]. Because everyone would put pressure on him [A - influenced by other people]. If you get an ice cream first then you won't be able to get a biscuit second [C – influence of previous event]. Because she likes to win a prize instead of losing [A]. Because Elle want to win a prize [A]. Because if she doesn't roll the dice hard enough she is most likely to get no prize [B]. Because Ellie is a strong name and it is more likely she will [A – influence of a person's name].
a) b)	Lack of independence – Probability has patterns It goes biscuit, biscuit, ice cream so next is going to be biscuit [A]. Because after ice cream goes biscuit [A].
a)	Availability – experience Students may be influenced by previous experiences. If they have recently found some money, they will think that it is far more likely to happen again than it is in reality. Because the first person doesn't usually get something good [A]. Because people hardly ever roll a 6 [A]. Because 1,2,3 is easier than 4,5,6 [B].
a) b)	Availability – belief Students may have an underlying conviction that one particular outcome is more likely. [An] ice cream is worth more money [B]. Because you cannot win a biscuit [B]. Because 6 is a lucky number [B]. Because the highest number usually has less of a chance of appearing [C]. Because you can't get a 6 on the 1st roll [C]. To roll a six is hard but you could roll a four easier [A]. Because people are more likely to not win a prize than having hope to win a prize [B]. Girls have more trouble than boys do so I think she would roll in between the 1 and 3 [B].

Next steps:

Each of these probability misconceptions above is common, even with adults.

There are three main ways we suggest can help students overcome their incorrect ideas.

Have discussions between students. Get them to justify their own explanations and to critique the explanations of others. Click on the link Mathematical classroom discourse for more ideas on how to do this.
Have plenty of practical experiences of probability, and record the results of this. Increasing exposure to and experience of probability builds up an intuitive feel for chance.
Find ways to quantify probability. This could be with simple counts for fixed sample sizes (up to Year 6), or as a fraction, decimal or percentage when the sample size varies (at Years 7 and 8).

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