6.3 Probability, tree diagrams and decision trees 243
In order to combine these, we calculate her
expected average interest. This average is
calculated as if she made a large number of
investments over a period of time with the
probabilities shown above: 60% of the time
she would earn $459 interest, and so on. Thus
the expected average amount she has at the
end of two years (remembering to add the
first and second years together) is:
60% of $5459 + 20% of $5618 + 20% of $5724
= $3275.40 + $1123.60 + $1144.80
= $5543.80This is a better option than the fixed interest
rate at $5512.50, but she would stand a 60%
risk of only having $5459.
In the following activity some of the
probabilities may seem quite arbitrary and
approximate, and the situation is rather
simplified, but real problems can often be
analysed usefully in this way. This is also
much more difficult in that it involves a
comparison of two probability trees with extra
added factors.
There are two ways I can go to work, both of
which involve a two-part journey. I can cycle to
the bus stop; this takes me 5 minutes
normally, or 15 minutes if a level crossing for
trains is closed on the way, which happens on
10% of occasions. A bus takes on average 5
minutes to come. I catch the first bus, which
may be a slow bus which takes 30 minutes or
a fast bus which takes 15 minutes. I get the
slow bus 20% of the time.
Alternatively, I can drive to the Park and
Ride car park. Driving usually takes me 15
minutes, but about half the time there is a
traffic jam and it takes 20 minutes. When I
get to the Park and Ride, I sometimes get the
bus straight away, but 60% of the time I have
to wait 10 minutes for the next one. The bus
takes 10 minutes to get me to work.
1 What is my shortest time to get to work?
2 On average, what is my best option for
getting to work and how long will it take me?
3 What are the chances of the first journey
option taking 40 minutes or more?Activity