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.

  1. What is the probability that you will draw a King?
  2. What are the odds that you will draw a King?
  3. What is the probability that you will draw a picture card (Ace, King, Queen or Jack)?
  4. What are the odds that you will draw a picture card (Ace, King, Queen or Jack)?
  5. Suppose I draw two cards from the pack. Are the events ‘Draw a King first’ and ‘Draw a Queen second’ independent events?
  6. 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:

… 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…

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.)

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).

  1. What is the probability that a randomly chosen death in 1997 was of a pilot 50 or older?
  2. What proportion of deaths from 1997 to 1999 were of pilots aged under 30?
  3. What other information may be useful in studying the effect of age on pilot deaths?
TABLE 16.2: Aviation deaths of private pilots in Australia from 1997 to 1999 according to the pilot’s age
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.

  1. ‘Whether or not I walk to work tomorrow morning,’ and ‘Whether or not rain is expected tomorrow morning.’
  2. ‘Whether or not a person smokes more than 10 cigarettes per week on average’ and ‘Whether or not a person get lung cancer.’
  3. ‘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.

  1. Suppose 100 people who do have a disease are tested. How many would be expected to return a positive test result?
  2. 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:

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.

The reasoning is incorrect. Explain why.

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.

Do you think the selection process really is ‘at random?’ Explain.