Thinking Skills: Critical Thinking and Problem Solving

Answers to assignments 337

If we throw far, near, far, the probabilities

of throwing two in a row are as follows:

Hit, hit, miss:^13 ×^12 ×^23 =^218

Miss, hit, hit:^23 ×^12 ×^13 =^218

Hit, hit, hit:^13 ×^12 ×^13 =^118

The total probability of winning is^518

or about 28%. The second strategy is

better. Some may regard this as counter-

intuitive as it involves two throws at

the harder target. Did you expect this

answer? Can you rationalise why the

second strategy should be the best?

Can you prove that it works for all

probabilities (as long as the farther target

is harder to hit)?

4    We first need to do some calculations

on the various options. These are

summarised in the table on page 338,

with the second column showing the

figures for a machine achieving a 99%

detection rate and the third column

showing those for a machine achieving

a 95% detection rate. Fixed costs are

ignored; these figures just represent the

total income minus the quality control

costs for the different assumptions.

We can now construct the decision tree,

as shown below.

The differences are quite small – the

present system shows a saving of $830

in almost $1 million. However, the

automatic system carries an 80% chance

of the loss being $1750.

6.4 Have you solved it?

Variable responses

7.1 Conditions and conditionals

1    a Reading the book is a necessary but

not a sufficient condition for passing

the exam.

Automatic system

Stay with manual QC system

Detection rate 99%

Detection rate 95%

20% chance

80% chance

$936,750

$941,350

$938,500

Income Contribution

to expected

value

$188,270

$749,400

Overall expected value $937,670