A man went to Monte Carlo to try and make his fortune. Whilst he
was there he had an opportunity to bet on the outcome of rolling
dice. He was offered the same odds for each of the following
outcomes: At least 1 six with 6 dice. At least 2 sixes with 12
dice. At least 3 sixes with 18 dice.
Two bags contain different numbers of red and blue balls. A ball is
removed from one of the bags. The ball is blue. What is the
probability that it was removed from bag A?
You and I play a game involving successive throws of a fair coin.
Suppose I pick HH and you pick TH. The coin is thrown repeatedly
until we see either two heads in a row (I win) or a tail followed
by a head (you win). What is the probability that you win?
Practical work seems particularly important for this
Start with estimating 10 cm. Give the group a minute or two to
practise with a ruler, then with all measures or samples out of
sight ask them to put two marks on a fresh piece of paper. The data
are collected, to the nearest mm. Before examining the
data invite students to make a guessed description of the data set.
Is it symmetric ? What is it centred on? How dispersed?
Next take five dice and run 20 trials counting the number of
sixes each time. Ask students to plot that frequency
If possible have each member of the group with a shuffled pack
of cards. Conduct the synchronised card turning and collect the
number of Aces of Spades observed at each card turn. If the group
is small perhaps report every ace, and later ask what effect this
change in the rules had on the distribution. Similarly any royal
card might also be included in the set of cards reported. As before
ask the group how this affects the distribution.
Now work with tossed coins. Five, as with the dice, and also one
coin for every member of the group. This should help students see
how the dice and cards activities are structurally the same. They
differ from the coin tossing in their asymmetry and it can be seen
how both the chance of a sighting (six or ace) and the sample size
(five dice or hundred packs of cards) affect the
Although the vocabulary 'discrete' and 'continuous' may help
distinguish these two from the distribution of 10cm estimates,
acquisition of correct technical terms is not the most important
There is plenty to discuss about the 10cm estimates: Is this
variable random? Is it symmetric? Does this data sample match
One key point to include within the discussion is that once we
collect the data to the nearest mm it becomes discrete but the
variable itself was continuous (there are no two distinct values
which cannot have another value between them)
How do we compare two sets of data, say for height statistics
for 11 year olds now and fifty years ago? Why can't we do exactly
the same with probability and sample data?
When five dice are rolled what is the probability that we see no
sixes, or one six only, or two sixes, three, four, or five? What
will the probability values for each of these come to as a
When a person estimates 10cm do you think there is probably more
chance of their estimate being within one centimetre of 10cm than
of being off by between 5cm and 6cm? Do you think estimating too
much is as likely as too little? Why?
How is Aces High like Five Dice? How is it different? And what
if this involved tossing a coin rather than dice and cards?
10cm is different to Five Dice in some key ways - what would you
say those differences were?