don steward
mathematics teaching 10 ~ 16

Saturday, 29 April 2017

two reds unit fraction

combined events, without replacement

two consecutive numbers of red and white counters

two reds fair game

it might be interesting to try to devise a game where the probability of obtaining two red balls (in two picks, without replacement) is 1/2

how many red balls and how many yellow balls would you need?

there is one easy to obtain answer and then lots of difficult ones...

two dice experiments

considering whether a 'game' is fair or not
using a sample space diagram

the powerpoint

theoretical outcomes

NRich have developed simulations for two of Geoff Giles' DIME material probability experiments (no longer available unfortunately)

trials can be simulated and then compared with a theoretical analysis

I think a theoretical consideration is difficult
a teacher could present an argument and then see if students can recreate it

the powerpoint

two dice and generalising

starting off looking at how many times a 4, 5 or 6 is obtained on any or both of two dice

identifying theoretical probabilities from a sample space and comparing this with the experimental data: student results having been collected together

exploring further, developing a general rule

the powerpoint

thanks to Cheadle Hulme High School for letting me teach this lesson and their subsequent insights

Friday, 28 April 2017

fair or not?

sample space diagrams provide a way to decide whether a game is fair or not

combined probability diagrams

there are various diagrams that can be used to help calculate combined probabilities

I suggest a progression:
  • array
  • route
  • frequency tree
  • probability tree 

with replacement powerpoint

without replacement powerpoint

balls in a bag

from a problem posed by David Wells

possibly of more interest as a context for multiplying fractions (involving cancelling) than combined event probability (without replacement)

note that the unitary fractions 1/4, 1/9, 1/16 ... are impossible to achieve

Saturday, 22 April 2017

geometric sequences

a revamped version of a previous posting, this one without surds

quadratic nth term