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Probability |
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If there is a 50-50 chance that something can
go wrong, |
1. 30 percent of all
passengers who fly from Vancouver to Toronto fly on White Knuckle Air.
Airlines misplace luggage for 24% of their passengers; 95% of this lost luggage
is subsequently recovered (within a year or two).
a. Fill the probabilities
into the following table.
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Vancouver to Toronto |
Luggage handling |
Luggage recovery |
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White Knuckle Air |
Other Carriers |
Luggage arrives |
Luggage |
Luggage recovered |
Luggage not recovered |
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b. If a passenger who has
flown from Vancouver to Toronto is randomly selected,
what is the probability that the selected individual flew on White Knuckle Air (event A), had luggage
misplaced (event B), and subsequently recovered the misplaced baggage (event
C)?
c. What if we select
another passenger at random, this time we are interested in the probability
that they did not fly with White
Knuckle Air, still had their luggage misplaced, but never recovered it?
2. In coastal towns in the Pacific Rim region, there is a
serious risk of flooding by tsunamis (large waves caused by the vibrations of
earthquakes or eruptions). Tsunamis travel quickly and are often very destructive.
Therefore, warning systems with sirens have been installed so people can
evacuate to higher ground in time. When the system is tested, individual sirens
sometimes fail. The sirens operate independently of one another (i.e. the
failure of one siren does not change the probability that any other siren
fails). Suppose that you live on the coast and two sirens can be heard from
your home:
a. If the system is activated, and there is a 3%
chance of any siren failing, what is the probability that two sirens will fail?
(Check your decimals J )
b. Suppose that a tsunami over 9.2 metres in height
will destroy most towns along the outer BC coast. If the mean height of
recorded tsunamis is 4.5 metres with a standard deviation of 2.1 metres,
calculate the probability that a
i) Tsunami wave will be
over 9.2 metres
ii) Tsunami wave will be
over 9.2 metres
(Hint: You must use Z values to answer part b)
3. A large corporation has 1200 employees - 960 men and 240 women. Over the past two years, 324 employees been promoted. The breakdown for males and females is shown below.
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Male |
Female |
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Promoted |
288 |
36 |
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Not Promoted |
672 |
204 |
After reviewing the data, a labour equity committee charged discrimination on the basis that 288 males had been promoted and only 36 women. Management countered that the relatively low number of promotions for women was not due to discrimination but due to the fact that there are fewer women in the workforce. Although probabilities cannot prove discrimination, they can lend support to an argument.
a. Convert the raw data into a joint probability table.
b. Calculate probability that a randomly selected person is promoted.
c. Calculate the probability of promotion given that the employee is male.
d. Calculate the probability of promotion given that the employee is female.
e. Briefly summarize
your analysis of the probabilities and your conclusions in a paragraph.
4. A hydrologist participating in the
Northland Drainage Basin Survey in New
Zealand collected information
on rainfall (mean annual) and the amount of runoff (mean annual specific
discharge). The data are available in the file (Rain.sav).
Use the Z scores and the Z distribution
to answer these questions. (Sketches needed for full marks on b,c,d,e,f)
a. Calculate the mean and standard deviation
for both variables (use
b. What is the probability of mean annual
rainfall in the study area being between 1,440 and 1,582 mm?
c. What is the probability of mean annual
specific discharge (runoff) in the study area exceeding 1,150 cubic metres per
second?
d. Based on the data, estimate the mean
annual specific discharge for a year when there is a one in a 100 year flood
event.
e. Drought conditions in the Northland
region are defined as a year where mean annual rainfall is below 1,400 mm. In
an adjacent watershed, the mean annual rainfall is 1,650 mm with a standard
deviation of 232 mm. Assuming the rainfall in the
adjacent watershed follows a normal distribution; determine the probability of
a drought in a given year.
f. The researcher refers to rainfall records for the last
50 years and calculates that the actual probability of a drought is 0.18 in
this watershed. Based on this information, do you expect the distribution of
rainfall in this watershed to be negatively skewed or positively skewed?
Explain your answer.
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Show all lab work, including tables and calculations! |
Marking Guide (Lab Total = 36)
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Q1 a |
3 |
Q4 a |
2 |
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Q1 b |
1 |
Q4 b |
3 |
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Q1 c |
1 |
Q4 c |
3 |
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Q2 a |
2 |
Q4 d |
3 |
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Q2 b |
i) 3 |
Q4 e |
3 |
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Q2 b |
ii)1 |
Q4 f |
2 |
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Q3 a |
4 |
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Q3 b |
1 |
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Q3 c |
1 |
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Q3 d |
1 |
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Q3 e |
2 |
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