240 Unit 6 Problem solving: further techniques
6.3 Probability, tree diagrams and decision trees
decision trees
6.3
get the overall chance of this combination:(^47) × (^36) = (^27).
However, we might get a blue ball first with
probability^37. The chances of then drawing a
red ball second are^46 , so the overall probability
is^46 ×^37 =^27 as before. The overall probability
of drawing red/blue in either order is the sum
of these, i.e.^47.
This problem could also have been solved
using a tree diagram (see the next page),
although in this case it would have required
more calculation.
The activity below is a probability problem
with a slight twist which takes it beyond being
a simple mathematical calculation.
At a village fair there is a game of chance
that involves throwing two dice. The dice are
normal, numbered 1 to 6. One is red and one
is blue. The number on the red die is
multiplied by 10 and added to the number on
the blue die to give a two-digit number. (So, if
red is 2 and blue is 4, your score is 24.) You
win a prize if you score more than 42.
What are the chances of winning?
Activity
Commentary
There are 36 (6 × 6) possible throws in all. If
the red die shows 1, 2 or 3, whatever the blue
die shows, you lose (18 of the throws). If the
red die shows 5 or 6, whatever the blue die
shows, you win (12 of the throws). This leaves
6 possible throws with the red die showing 4:
Simple probability
Questions involving probability can occur at
all levels of thinking skills examinations. In
AS Level examinations, these are usually
restricted to simple probability (e.g. the
chances of a 6 coming up in a single throw of
a die) or direct combinations of two
probabilities (e.g. the chances of the numbers
on two dice adding to 7). In the latter case, we
need to distinguish between the combinations
being dependent on each other or
independent. The sum of the numbers on two
dice is an example of an independent
combination – one die is not affected by the
other, and each can randomly show any
number from 1 to 6.
An example of a dependent combination,
where one operation depends on the results of
another, is the drawing of coloured balls from
a bag without replacement.
A bag contains four red balls and three blue
balls. If two balls are removed from the bag,
what are the chances of drawing one red and
one blue ball?
Activity
Commentary
We must look at all the possibilities. The
chances of drawing a red ball first are^47. The
chances of then drawing a blue ball are^36 (not
(^37) as we have already taken one ball out). We
can then multiply the probabilities together to