## 16.8 Exercises

Selected answers are available in Sect. D.16.

**Exercise 16.1 **Suppose you have a well-shuffled, standard pack of 52 cards.

- What is the
*probability*that you will draw a**King**? - What are the
*odds*that you will draw a**King**? - What is the
*probability*that you will draw a picture card (**Ace**,**King**,**Queen**or**Jack**)? - What are the
*odds*that you will draw a picture card (**Ace**,**King**,**Queen**or**Jack**)? - Suppose I draw two cards from the pack.
Are the events ‘Draw a
**King**first’ and ‘Draw a**Queen**second’ independent events? - Suppose I draw one card from the pack and roll a six-sided die.
Are the events ‘Draw a
**Jack**from the pack of cards’ and ‘Roll a**5**on the die’ independent events?

**Exercise 16.2 **On October 13,
the American television programme *Nightline* interviewed
Dr Richard Andrews, director of the California Office of Emergency Services.
They discussed various natural disasters that were being predicted as a result of an El Nino.
In the interview, Dr Andrews said:

Explain why Dr Andrews is incorrect when he says that “every probability is 50–50.” Give an example to show why he must be incorrect. (Based on a report in Chance News 6.12.)… we have to take these forecasts very seriously […] I listen to earth scientists talk about earthquake probabilities a lot and in my mind every probability is 50–50, either it will happen or it won’t happen…

**Exercise 16.3 **The data in Table 16.2
were obtained from an investigation into aviation deaths of private pilots in Australia (Ruscoe and Dunn 2003).

- What is the probability that a randomly chosen death in 1997 was of a pilot 50 or older?
- What proportion of deaths from 1997 to 1999 were of pilots aged under 30?
- What other information may be useful in studying the effect of age on pilot deaths?

1997 | 1998 | 1999 | |
---|---|---|---|

Under 30 | 2 | 1 | 3 |

30 to 49 | 5 | 12 | 5 |

50 or over | 9 | 11 | 9 |

**Exercise 16.4 **Are these pairs of two events likely to be *independent* or *not independent*? Explain.

- ‘Whether or not I walk to work tomorrow morning,’ and ‘Whether or not rain is expected tomorrow morning.’
- ‘Whether or not a person smokes more than 10 cigarettes per week on average’ and ‘Whether or not a person get lung cancer.’
- ‘Whether or not it rains today’ and ‘Whether or not my rubbish is collected today.’

**Exercise 16.5 **
In disease testing, two keys aspects of the test are:

- Sensitivity
*: the probability of getting a*positive* test result among people*who do*have the disease; and *Specificity*: the probability of getting a*negative*test result among people*who do not*have the disease.

Both are important for understanding how well a test works in practice.
Consider a test with a *sensitivity* of 0.99 and a *specificity* of 0.98.

- Suppose 100 people who
*do*have a disease are tested. How many would be expected to return a positive test result? - Suppose 100 people who
*do not*have a disease are tested. How many would be expected to return a positive test result?

**Exercise 16.6 **Consider the following argument:

The reasoning is incorrect. Explain why.When I toss two coins, there are only three outcomes: a Head and a Head, a Tail and a Tail, or one of each. So the probability of obtaining two Tails must be one-third.

**Exercise 16.7 **Since my wife and I have been married,
I have been called to jury service three times.
The latest notice reads:

Your name has been selected at random from the electoral role…

In the same length of time,
my wife has *never* been called to jury service.