A collection of memorable mathematics lessons that conference and colloquia delegates had observed or taught which they felt were successful. Each account refers to one or more of the values and principles in the report.

Lesson Account 26 - Expected Values

Written by

Ken McKelvie

Organisation

University of Liverpool

Age/Ability Range

Talk to University of Liverpool-run Maths Club. 15-18 years. Duration 70 mins. Topic - Expected Values

How was the session/task introduced? Practical coin spinning - participants working in pairs with one spinning and one recording the observed results. Calculations of cumulative proportions of Heads obtained were made. Counts of different “run” lengths of Heads were noted. Calculations of average “run” lengths were made.

How was the session/task sustained? Intuitive concept of “expected values” in context of results recorded was established.

The notion of discrete probability distributions was introduced.

A formal definition of an “expected value” was given.

Exercises to consolidate participants’ understanding of the given formal definition were completed – with individual help and guidance where required.

The aim of deriving a calculated expected value for run lengths motivated the derivation of some standard results for the summation of geometric and related series.

The various relevant interpretations of subsets of entries in Pascal’s Triangle were noted. The appropriate probability distribution for run length was derived. The sum of the probabilities was verified as 1 (as anticipated) and the corresponding expected value was confirmed to be 2 (as anticipated).

How was the session/task concluded? As a light-hearted finale: Participants were invited to spin their coin repeatedly and, at each spin, to score +1 for a Head or -1 for a Tail, and, also at each spin, to calculate the cumulative sum of these scores. The number of spins taken to the cumulative sum’s first ‘return to zero’ was to be recorded.

After a few minutes, several “returns to zero” were noted but several participants had cumulative sums that were drifting away from zero.

At this point, the presenter suggested that it might make a party game for participants’ friends, revealing (but not proving!) the sting in the tail. That is that whilst the probability of return to zero is 1, the expected number of spins required to the first return to zero is “infinite”.

Participants appeared to enjoy the fact that some were successful in completion, others were not, and the explanation as to why.

What were the critical moments? Establishing notion of discrete probability distribution. Intuitive concept of expected value.

Acceptance of notion of summation of series.

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning? Interpretations of Pascal’s Triangle.

How was that mathematics learnt? Experiment and discussion. Some practice for consolidation

Other memorable outcomes No answer

Resources Available on website – University of Liverpool - Dep’t of Mathematical Sciences –