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Guide and features
Guide and features
Science, Technology, Engineering and Mathematics
Featured Early Years Foundation Stage; US Kindergarten
Featured UK Key Stage 1&2; US Grades 1-4
Featured UK Key Stage 3-5; US Grades 5-12
Featured UK Key Stage 1, US Grade 1 & 2
Featured UK Key Stage 2; US Grade 3 & 4
Featured UK Key Stages 3 & 4; US Grade 5-10
Featured UK Key Stage 4 & 5; US Grade 11 & 12
This problem makes use of the
Why do this problem?
This problem requires students to invent a distribution to satisfy certain criteria. In doing so, they will strongly reinforce their understanding of the meaning of distributions as a whole. Students will have to think creatively and really engage with the relationship between a distribution and the probabilities in order to fully solve the problem. The fact that universal order exists underlying all random processes (i.e. the inequalities) is interesting; instead of a collection of diverse results, distributions become part of a coherent whole.
A main part of this challenge is understanding the meaning of the question. Spend time with the group exploring what they feel that question is asking and encouraging them to work through the meaning of the parts of the question. Once they have a feel for the question they might want to proceed by:
Looking at the inequalities for some known distributions (such as the binomial or the normal)
Investigating other simple probability distributions (such as that obtained on the roll of a die)
Creating distributions 'randomly' using the distribution maker and investigating the inequalities for the probability distributions obtained.
The distribution maker interactivity will help students to see probabilities for different distribution results.
What is the problem asking?
How can we use the condition that (??) is a whole number?
Are then any obvious distributions to try out?
As a challenge, try to create a distribution which violates the inequalities. How close can you get? Does this give you a feel for why the inequalities hold? How close to the inequalities do distributions which the class already know about (such as binomial or normal) get?
You might like to refer keen students to the
central limit theorem
in probability, which shows that on average distributions tend to the normal.
Suggest for the first part that students tabulate the probability that X exceeds 1, 2, 3, 4, 5 and 6 for the roll of a single die. Repeat for the roll of 2 dice (these situations are pre-programmed into the distribution maker). Using these explicit results they can explore the inequalities.
Probability distributions, expectation and variance
Manipulating algebraic expressions/formulae
Mathematical reasoning & proof
Maths Supporting SET
Meet the team
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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NRICH is part of the family of activities in the
Millennium Mathematics Project